Trigonometry

Problem 1901

Calculate the value of $1 - \sin^2 30^{\circ} - \sin^2 60^{\circ}$ without a calculator. Options: $\frac{1-\sqrt{3}}{2}$ , 1, 0, $\frac{1}{4}$ .

See Solution

Problem 1902

Find $\cot \theta$ if $\cos \theta=\frac{3 \sqrt{10}}{10}$ .

See Solution

Problem 1903

Is the value of $\sin \left(\frac{\pi}{12}\right) = \frac{\sqrt{2-\sqrt{3}}}{2}$ exact or approximate? Explain.

See Solution

Problem 1904

Calculate the value of $\tan \frac{\pi}{6} - \sin \frac{\pi}{3}$ without a calculator.

See Solution

Problem 1905

Solve for $0^{\circ} \leq \theta < 360^{\circ}$ where $\cos \theta = \frac{1}{2}$ . What are the values of $\theta$ ?

See Solution

Problem 1906

Calculate the value of $\tan \frac{\pi}{6} - \sin \frac{\pi}{3}$ without a calculator.

See Solution

Problem 1907

Find the exact value of $\csc \theta$ given $\sin \theta=\frac{1}{4}$ and $\cos \theta=\frac{\sqrt{15}}{4}$ .

See Solution

Problem 1908

Find the value of $\cot A$ in triangle $ABC$ where $b=5$ and $c=6$ . Provide exact answers with rational denominators.

See Solution

Problem 1909

Find the length of a guy wire attached 10 ft from the top of a 230 ft tower at a $32^{\circ}$ angle. Round to the nearest tenth.

See Solution

Problem 1910

Find the radian measure of a $300^{\circ}$ angle: $\frac{3 \pi}{5}$ , $\frac{5 \pi}{3}$ , $2 \pi$ , $150 \pi$ .

See Solution

Problem 1911

Find the distance between two cars below a 1000-foot cliff with angles of depression $21^{\circ}$ and $28^{\circ}$ .

See Solution

Problem 1912

Find the expression equivalent to $\frac{\tan x-1}{\tan x+1}$ using identities. Options include:
1. $\frac{\cot x}{1-\cot x}$
2. $\frac{\cot x}{1+\cot x}$
3. $\frac{1-\cot x}{1+\cot x}$
4. $\frac{1+\cot x}{1-\cot x}$

See Solution

Problem 1913

Find the exact value of $\tan \frac{\pi}{4}$ . Options: $\frac{\sqrt{3}}{3}$ , $-1$ , 0, 1.

See Solution

Problem 1914

Find the exact value of $\tan (30 \pi)$ using a coterminal angle. Options: -1, 0, 1, undefined.

See Solution

Problem 1915

Find $\sin \theta$ for the point $(4,-3)$ . Options: $-3$ , $\frac{4}{5}$ , $-\frac{3}{5}$ , $\frac{5}{4}$ .

See Solution

Problem 1916

Find $\csc B$ for a right triangle with sides $a=5$ and $b=6$ . Provide exact answers with rational denominators.

See Solution

Problem 1917

Find the exact value of $-\frac{\sec 10^{\circ}}{\csc 50^{\circ}}$ using identities. Choices: $-1$ , $1$ , $0$ , undefined.

See Solution

Problem 1918

Find the exact value of $\tan(-690^{\circ})$ using a coterminal angle, without a calculator. Choices: $\frac{\sqrt{3}}{3}$ , $\sqrt{3}$ , $-\sqrt{3}$ , $\frac{\sqrt{3}}{2}$ .

See Solution

Problem 1919

Given point $(-3,-4)$ , find $\sec \theta$ . Options: $-\frac{3}{5}$ , $-\frac{5}{3}$ , $\frac{5}{4}$ , $\frac{4}{3}$ .

See Solution

Problem 1920

Find the exact value of $\sin \frac{5 \pi}{3}$ using the reference angle, without a calculator. Options: $-\frac{1}{2}$ , $\frac{\sqrt{3}}{2}$ , $-1$ , $-\frac{\sqrt{3}}{2}$ .

See Solution

Problem 1921

Find $f(45^{\circ})$ for $f(\theta)=\sin \theta$ . What is the value? Options: $\frac{\sqrt{2}}{2}$ , $\frac{1}{2}$ , $\sqrt{2}$ , $-\frac{\sqrt{2}}{2}$ .

See Solution

Problem 1922

Find $\sec \theta$ for the point $P(-3,-1)$ on the circle $x^{2}+y^{2}=r^{2}$ .

See Solution

Problem 1923

Find $\theta$ in $[0^{\circ}, 90^{\circ}]$ such that $\tan \theta = 0.75248493$ . Calculate $\theta \approx$ .

See Solution

Problem 1924

Is the statement true or false? Evaluate if $1+\tan^{2} 30.1^{\circ} = -\sec^{2} 30.1^{\circ}$ .

See Solution

Problem 1925

Find the range of the tangent function: all reals except odd multiples of $\frac{\pi}{2}$ , or other options?

See Solution

Problem 1926

Determine the quadrant for angle $\theta$ where $\sin \theta > 0$ and $\cos \theta > 0$ .

See Solution

Problem 1927

If $\sin \theta=0.4$ , what is $\sin (\theta+\pi)$ ? Options: 0.4, -0.4, -0.6, 0.6.

See Solution

Problem 1928

Find the reference angle for the angle $-\frac{42 \pi}{8}$ .

See Solution

Problem 1929

Find $\tan \theta$ for the point $P(5,4)$ on the circle $x^{2}+y^{2}=r^{2}$ .

See Solution

Problem 1930

Find the exact value of $\sec \frac{3 \pi}{4}$ using the reference angle, without a calculator.

See Solution

Problem 1931

Find the reference angle for $-\frac{42 \pi}{8}$ . Options: $\frac{\pi}{4}$ , $\frac{\pi}{2}$ , $\frac{\pi}{3}$ , $\frac{\pi}{8}$ .

See Solution

Problem 1932

Find $\sin \theta$ for the point $P(-4,-3)$ on the circle $x^{2}+y^{2}=r^{2}$ . Choices: $-\frac{3}{5}$ , $\frac{3}{5}$ , $\frac{4}{5}$ , $-\frac{4}{5}$ .

See Solution

Problem 1933

Find $\cot \theta$ if $\cos \theta = \frac{21}{29}$ and $\frac{3 \pi}{2} < \theta < 2 \pi$ .

See Solution

Problem 1934

Find $\sin t$ for the point $P\left(-\frac{\sqrt{77}}{9},-\frac{2}{9}\right)$ on the unit circle.

See Solution

Problem 1935

Find $\tan t$ for the point $\mathrm{P} = \left(\frac{3}{8}, \frac{\sqrt{55}}{8}\right)$ on the unit circle.

See Solution

Problem 1936

Determine the quadrant for angle $\theta$ given that $\cos \theta > 0$ and $\csc \theta < 0$ .

See Solution

Problem 1937

Determine the quadrant for angle $\theta$ if $\csc \theta>0$ and $\sec \theta>0$ .

See Solution

Problem 1938

What is the range of the secant function? Options: 1) [-1, 1] 2) all reals except odd multiples of $\frac{\pi}{2}$ 3) $\geq 1$ or $\leq -1$ 4) all reals

See Solution

Problem 1939

Find the exact value of $\tan \frac{-3 \pi}{4}$ using the reference angle without a calculator.

See Solution

Problem 1940

Find the exact value of $\cot \frac{-11 \pi}{6}$ using the reference angle without a calculator.

See Solution

Problem 1941

Determine the quadrant for angle $\theta$ given that $\cot \theta<0$ and $\cos \theta>0$ .

See Solution

Problem 1942

What is the range of the sine function: $[-1, 1]$ ?

See Solution

Problem 1943

Determine the quadrant for angle $\theta$ if $\sec \theta<0$ and $\tan \theta<0$ .

See Solution

Problem 1944

Find the values of $\theta$ for which the function $f(\theta)=\sec \theta$ is undefined.

See Solution

Problem 1945

Find the values of $\theta$ for which $f(\theta)=\sec \theta$ is undefined.

See Solution

Problem 1946

Determine the quadrant for angle $\theta$ given that $\sin \theta > 0$ and $\cos \theta < 0$ .

See Solution

Problem 1947

Determine the quadrant for angle $\theta$ given that $\tan \theta>0$ and $\sin \theta<0$ .

See Solution

Problem 1948

Find $\sec \theta$ if $\sin \theta = -\frac{4}{9}$ and $\tan \theta > 0$ .

See Solution

Problem 1949

Find $\cos \theta$ and $\tan \theta$ given $\sin \theta=\frac{1}{2}$ and $\sec \theta<0$ .

See Solution

Problem 1950

Find the reference angle for $-\frac{5 \pi}{6}$ . Options: $\frac{\pi}{12}$ , $\frac{\pi}{6}$ , $\frac{7 \pi}{6}$ , $\frac{5 \pi}{6}$ .

See Solution

Problem 1951

Determine the quadrant for angle $\theta$ given that $\cot \theta>0$ and $\sin \theta<0$ .

See Solution

Problem 1952

Find $\cos \theta$ if $\tan \theta = -\frac{10}{7}$ and $\theta$ is in quadrant II.

See Solution

Problem 1953

In a right triangle with hypotenuse 100km and opposite side 60km, find $\sin \theta = \frac{60}{100}$ .

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Problem 1954

Find the quadrant for angle $\theta$ where $\tan \theta<0$ and $\sin \theta<0$ .

See Solution

Problem 1955

Determine the quadrant for angle $\theta$ given that $\cos \theta < 0$ and $\csc \theta < 0$ .

See Solution

Problem 1956

Find $\tan \theta$ given $\sec \theta = \frac{3}{2}$ and $\theta$ is in quadrant IV.

See Solution

Problem 1957

Find $\csc \theta$ given that $\sin \theta = \frac{1}{3}$ .

See Solution

Problem 1958

Find the reference angle for the angle $\frac{7 \pi}{8}$ .

See Solution

Problem 1959

Find $\cos \theta$ for the point $P(12, 5)$ on the circle $x^{2}+y^{2}=r^{2}$ .

See Solution

Problem 1960

Find $\cot \theta$ for the point $P(-3,2)$ on the circle $x^{2}+y^{2}=r^{2}$ .

See Solution

Problem 1961

Find $\cos \theta$ if $\tan \theta=-\frac{15}{8}$ and $90^{\circ}<\theta<180^{\circ}$ .

See Solution

Problem 1962

Find the sine of angle $t$ for the point $P\left(\frac{5}{8}, \frac{\sqrt{39}}{8}\right)$ on the unit circle.

See Solution

Problem 1963

Find the cosine of angle $t$ for the point $P\left(-\frac{\sqrt{55}}{8}, \frac{3}{8}\right)$ on the unit circle.

See Solution

Problem 1964

Find the domain of the sine function. What values can $x$ take for $\sin(x)$ ?

See Solution

Problem 1965

Find the values of $\theta$ for which $f(\theta)=\csc \theta$ is undefined.

See Solution

Problem 1966

Find the values of $\theta$ for which $f(\theta)=\csc \theta$ is undefined.

See Solution

Problem 1967

Find $\csc \theta$ given $\cot \theta = -\frac{9}{4}$ and $\cos \theta < 0$ .

See Solution

Problem 1968

Identify which trigonometric values are negative: I. $\sin \left(-292^{\circ}\right)$ , II. $\tan \left(-193^{\circ}\right)$ , III. $\cos \left(-207^{\circ}\right)$ , IV. $\cot 222^{\circ}$ .

See Solution

Problem 1969

Identify which trigonometric values are negative: I. $\sin \left(-292^{\circ}\right)$ II. $\tan \left(-193^{\circ}\right)$ III. $\cos \left(-207^{\circ}\right)$ IV. $\cot 222^{\circ}$ . Options: I and III, II and III, III only, II, III, and IV.

See Solution

Problem 1970

If $\tan \theta=a$ (where $a \neq 0$ ), determine $\cot \theta$ .

See Solution

Problem 1971

If $\tan \theta=a(a \neq 0)$ , what is $\cot \theta$ using reciprocal identities?

See Solution

Problem 1972

Find the reference angle for the angle $-\frac{13 \pi}{12}$ .

See Solution

Problem 1973

Find the reference angle for $\frac{4 \pi}{3}$ . Options: $\frac{2 \pi}{3}$ , $\frac{\pi}{6}$ , $\frac{\pi}{3}$ , $\frac{4 \pi}{3}$ .

See Solution

Problem 1974

Find $\csc \theta$ for the point $P(-3,-1)$ on the circle $x^{2}+y^{2}=r^{2}$ .

See Solution

Problem 1975

Find $\tan \theta$ using $\sin \theta=\frac{6 \sqrt{37}}{37}$ and $\cos \theta=\frac{\sqrt{37}}{37}$ .

See Solution

Problem 1976

If $\sin \theta=\frac{\sqrt{3}}{3}$ , calculate $(\sin \theta)^{2}$ .

See Solution

Problem 1977

If $\tan \theta=3$ , calculate $\tan^{3} \theta$ .

See Solution

Problem 1978

Find $\cot \theta$ using the values $\sin \theta=-\frac{9}{41}$ and $\cos \theta=-\frac{40}{41}$ .

See Solution

Problem 1979

Find $\tan \theta$ if $\sin \theta = \frac{8}{17}$ and $\cos \theta = \frac{15}{17}$ .

See Solution

Problem 1980

Find $\cos \theta$ given $\sin \theta=\frac{1}{2}$ and $\theta$ in QII. $\cos \theta=\square$

See Solution

Problem 1981

Find $\sin \theta$ given $\cos \theta = \frac{4}{5}$ and $\theta$ is in QI. $\sin \theta =$

See Solution

Problem 1982

Find $\cos \theta$ given $\sin \theta = -\frac{4}{5}$ and $\theta$ is in QIII. Use the first Pythagorean identity. $\cos \theta =$

See Solution

Problem 1983

Find $\sec \theta$ given that $\sin \theta = \frac{8}{17}$ and $\cos \theta = \frac{15}{17}$ .

See Solution

Problem 1984

Find $\sin \theta$ if $\cos \theta = \frac{1}{2}$ and $\theta$ is in quadrant I.

See Solution

Problem 1985

Find $\tan \theta$ given $\sin \theta = \frac{3}{4}$ and $\theta$ is in QI. What is $\tan \theta$ ?

See Solution

Problem 1986

Find $\sec \theta$ given $\tan \theta = \frac{9}{40}$ and $\theta$ is in QIII. $\sec \theta =$

See Solution

Problem 1987

Find $\csc \theta$ given $\cot \theta=-48/55$ and $\sin \theta>0$ . What is $\csc \theta$ ?

See Solution

Problem 1988

Express $\cos \theta$ using only $\sin \theta$ .

See Solution

Problem 1989

Express $\sec \theta$ using only $\cos \theta$ .

See Solution

Problem 1990

Find the trigonometric ratios for $\theta$ if $\sin \theta=-\frac{\sqrt{3}}{2}$ and $\theta$ is not in QIII. $\cos \theta=$ $\tan \theta=$ $\cot \theta=$ $\sec \theta=$ $\csc \theta=$

See Solution

Problem 1991

Rewrite $csc \theta$ using only $\cos \theta$ .

See Solution

Problem 1992

Simplify the expression: $\frac{\frac{1}{\cos \theta}}{\frac{1}{\sin \theta}}$ .

See Solution

Problem 1993

Rewrite $\csc \theta \cot \theta$ using $\sin \theta$ and $\cos \theta$ , then simplify if possible.

See Solution

Problem 1994

Rewrite $\csc \theta \tan \theta$ using $\sin \theta$ and $\cos \theta$ , then simplify if possible.

See Solution

Problem 1995

Rewrite $\frac{\sec \theta}{\csc \theta}$ using $\sin \theta$ and $\cos \theta$ , then simplify if possible.

See Solution

Problem 1996

Rewrite $\frac{\sec \theta}{\tan \theta}$ using $\sin \theta$ and $\cos \theta$ , then simplify.

See Solution

Problem 1997

Express $\frac{\tan \theta}{\cot \theta}$ using $\sin \theta$ and $\cos \theta$ , then simplify if possible.

See Solution

Problem 1998

Rewrite $\frac{\sin \theta}{\csc \theta}$ using $\sin \theta$ and $\cos \theta$ , and simplify if possible.

See Solution

Problem 1999

Express $\tan \theta + \sec \theta$ using $\sin \theta$ and $\cos \theta$ , then simplify if possible.

See Solution

Problem 2000

Rewrite $\sin \theta \cot \theta + 4 \cos \theta$ using $\sin \theta$ and $\cos \theta$ , then simplify.

See Solution

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Trigonometry

Problem 1901

Calculate the value of 1−sin⁡230∘−sin⁡260∘1 - \sin^2 30^{\circ} - \sin^2 60^{\circ}1−sin230∘−sin260∘ without a calculator. Options: 1−32\frac{1-\sqrt{3}}{2}21−3​​, 1, 0, 14\frac{1}{4}41​.

Problem 1902

Find cot⁡θ\cot \thetacotθ if cos⁡θ=31010\cos \theta=\frac{3 \sqrt{10}}{10}cosθ=10310​​.

Problem 1903

Is the value of sin⁡(π12)=2−32\sin \left(\frac{\pi}{12}\right) = \frac{\sqrt{2-\sqrt{3}}}{2}sin(12π​)=22−3​​​ exact or approximate? Explain.

Problem 1904

Calculate the value of tan⁡π6−sin⁡π3\tan \frac{\pi}{6} - \sin \frac{\pi}{3}tan6π​−sin3π​ without a calculator.

Problem 1905

Solve for 0∘≤θ<360∘0^{\circ} \leq \theta < 360^{\circ}0∘≤θ<360∘ where cos⁡θ=12\cos \theta = \frac{1}{2}cosθ=21​. What are the values of θ\thetaθ?

Problem 1906

Calculate the value of tan⁡π6−sin⁡π3\tan \frac{\pi}{6} - \sin \frac{\pi}{3}tan6π​−sin3π​ without a calculator.

Problem 1907

Find the exact value of csc⁡θ\csc \thetacscθ given sin⁡θ=14\sin \theta=\frac{1}{4}sinθ=41​ and cos⁡θ=154\cos \theta=\frac{\sqrt{15}}{4}cosθ=415​​.

Problem 1908

Find the value of cot⁡A\cot AcotA in triangle ABCABCABC where b=5b=5b=5 and c=6c=6c=6. Provide exact answers with rational denominators.

Problem 1909

Find the length of a guy wire attached 10 ft from the top of a 230 ft tower at a 32∘32^{\circ}32∘ angle. Round to the nearest tenth.

Problem 1910

Find the radian measure of a 300∘300^{\circ}300∘ angle: 3π5\frac{3 \pi}{5}53π​, 5π3\frac{5 \pi}{3}35π​, 2π2 \pi2π, 150π150 \pi150π.

Problem 1911

Find the distance between two cars below a 1000-foot cliff with angles of depression 21∘21^{\circ}21∘ and 28∘28^{\circ}28∘.

Problem 1912

Problem 1913

Find the exact value of tan⁡π4\tan \frac{\pi}{4}tan4π​. Options: 33\frac{\sqrt{3}}{3}33​​, −1-1−1, 0, 1.

Problem 1914

Find the exact value of tan⁡(30π)\tan (30 \pi)tan(30π) using a coterminal angle. Options: -1, 0, 1, undefined.

Problem 1915

Find sin⁡θ\sin \thetasinθ for the point (4,−3)(4,-3)(4,−3). Options: −3-3−3, 45\frac{4}{5}54​, −35-\frac{3}{5}−53​, 54\frac{5}{4}45​.

Problem 1916

Find csc⁡B\csc BcscB for a right triangle with sides a=5a=5a=5 and b=6b=6b=6. Provide exact answers with rational denominators.

Problem 1917

Find the exact value of −sec⁡10∘csc⁡50∘-\frac{\sec 10^{\circ}}{\csc 50^{\circ}}−csc50∘sec10∘​ using identities. Choices: −1-1−1, 111, 000, undefined.

Problem 1918

Find the exact value of tan⁡(−690∘)\tan(-690^{\circ})tan(−690∘) using a coterminal angle, without a calculator. Choices: 33\frac{\sqrt{3}}{3}33​​, 3\sqrt{3}3​, −3-\sqrt{3}−3​, 32\frac{\sqrt{3}}{2}23​​.

Problem 1919

Given point (−3,−4)(-3,-4)(−3,−4), find sec⁡θ\sec \thetasecθ. Options: −35-\frac{3}{5}−53​, −53-\frac{5}{3}−35​, 54\frac{5}{4}45​, 43\frac{4}{3}34​.

Problem 1920

Find the exact value of sin⁡5π3\sin \frac{5 \pi}{3}sin35π​ using the reference angle, without a calculator. Options: −12-\frac{1}{2}−21​, 32\frac{\sqrt{3}}{2}23​​, −1-1−1, −32-\frac{\sqrt{3}}{2}−23​​.

Problem 1921

Find f(45∘)f(45^{\circ})f(45∘) for f(θ)=sin⁡θf(\theta)=\sin \thetaf(θ)=sinθ. What is the value? Options: 22\frac{\sqrt{2}}{2}22​​, 12\frac{1}{2}21​, 2\sqrt{2}2​, −22-\frac{\sqrt{2}}{2}−22​​.

Problem 1922

Find sec⁡θ\sec \thetasecθ for the point P(−3,−1)P(-3,-1)P(−3,−1) on the circle x2+y2=r2x^{2}+y^{2}=r^{2}x2+y2=r2.

Problem 1923

Find θ\thetaθ in [0∘,90∘][0^{\circ}, 90^{\circ}][0∘,90∘] such that tan⁡θ=0.75248493\tan \theta = 0.75248493tanθ=0.75248493. Calculate θ≈\theta \approxθ≈.

Problem 1924

Is the statement true or false? Evaluate if 1+tan⁡230.1∘=−sec⁡230.1∘1+\tan^{2} 30.1^{\circ} = -\sec^{2} 30.1^{\circ}1+tan230.1∘=−sec230.1∘.

Problem 1925

Find the range of the tangent function: all reals except odd multiples of π2\frac{\pi}{2}2π​, or other options?

Problem 1926

Determine the quadrant for angle θ\thetaθ where sin⁡θ>0\sin \theta > 0sinθ>0 and cos⁡θ>0\cos \theta > 0cosθ>0.

Problem 1927

If sin⁡θ=0.4\sin \theta=0.4sinθ=0.4, what is sin⁡(θ+π)\sin (\theta+\pi)sin(θ+π)? Options: 0.4, -0.4, -0.6, 0.6.

Problem 1928

Find the reference angle for the angle −42π8-\frac{42 \pi}{8}−842π​.

Problem 1929

Find tan⁡θ\tan \thetatanθ for the point P(5,4)P(5,4)P(5,4) on the circle x2+y2=r2x^{2}+y^{2}=r^{2}x2+y2=r2.

Problem 1930

Find the exact value of sec⁡3π4\sec \frac{3 \pi}{4}sec43π​ using the reference angle, without a calculator.

Problem 1931

Find the reference angle for −42π8-\frac{42 \pi}{8}−842π​. Options: π4\frac{\pi}{4}4π​, π2\frac{\pi}{2}2π​, π3\frac{\pi}{3}3π​, π8\frac{\pi}{8}8π​.

Problem 1932

Find sin⁡θ\sin \thetasinθ for the point P(−4,−3)P(-4,-3)P(−4,−3) on the circle x2+y2=r2x^{2}+y^{2}=r^{2}x2+y2=r2. Choices: −35-\frac{3}{5}−53​, 35\frac{3}{5}53​, 45\frac{4}{5}54​, −45-\frac{4}{5}−54​.

Problem 1933

Find cot⁡θ\cot \thetacotθ if cos⁡θ=2129\cos \theta = \frac{21}{29}cosθ=2921​ and 3π2<θ<2π\frac{3 \pi}{2} < \theta < 2 \pi23π​<θ<2π.

Problem 1934

Find sin⁡t\sin tsint for the point P(−779,−29)P\left(-\frac{\sqrt{77}}{9},-\frac{2}{9}\right)P(−977​​,−92​) on the unit circle.

Problem 1935

Find tan⁡t\tan ttant for the point P=(38,558)\mathrm{P} = \left(\frac{3}{8}, \frac{\sqrt{55}}{8}\right)P=(83​,855​​) on the unit circle.

Problem 1936

Determine the quadrant for angle θ\thetaθ given that cos⁡θ>0\cos \theta > 0cosθ>0 and csc⁡θ<0\csc \theta < 0cscθ<0.

Problem 1937

Determine the quadrant for angle θ\thetaθ if csc⁡θ>0\csc \theta>0cscθ>0 and sec⁡θ>0\sec \theta>0secθ>0.

Problem 1938

What is the range of the secant function? Options: 1) [-1, 1] 2) all reals except odd multiples of π2\frac{\pi}{2}2π​ 3) ≥1\geq 1≥1 or ≤−1\leq -1≤−1 4) all reals

Problem 1939

Find the exact value of tan⁡−3π4\tan \frac{-3 \pi}{4}tan4−3π​ using the reference angle without a calculator.

Problem 1940

Find the exact value of cot⁡−11π6\cot \frac{-11 \pi}{6}cot6−11π​ using the reference angle without a calculator.

Calculate the value of $1 - \sin^2 30^{\circ} - \sin^2 60^{\circ}$ without a calculator. Options: $\frac{1-\sqrt{3}}{2}$ , 1, 0, $\frac{1}{4}$ .

Find $\cot \theta$ if $\cos \theta=\frac{3 \sqrt{10}}{10}$ .

Is the value of $\sin \left(\frac{\pi}{12}\right) = \frac{\sqrt{2-\sqrt{3}}}{2}$ exact or approximate? Explain.

Calculate the value of $\tan \frac{\pi}{6} - \sin \frac{\pi}{3}$ without a calculator.

Solve for $0^{\circ} \leq \theta < 360^{\circ}$ where $\cos \theta = \frac{1}{2}$ . What are the values of $\theta$ ?

Calculate the value of $\tan \frac{\pi}{6} - \sin \frac{\pi}{3}$ without a calculator.

Find the exact value of $\csc \theta$ given $\sin \theta=\frac{1}{4}$ and $\cos \theta=\frac{\sqrt{15}}{4}$ .

Find the value of $\cot A$ in triangle $ABC$ where $b=5$ and $c=6$ . Provide exact answers with rational denominators.

Find the length of a guy wire attached 10 ft from the top of a 230 ft tower at a $32^{\circ}$ angle. Round to the nearest tenth.

Find the radian measure of a $300^{\circ}$ angle: $\frac{3 \pi}{5}$ , $\frac{5 \pi}{3}$ , $2 \pi$ , $150 \pi$ .

Find the distance between two cars below a 1000-foot cliff with angles of depression $21^{\circ}$ and $28^{\circ}$ .

Find the exact value of $\tan \frac{\pi}{4}$ . Options: $\frac{\sqrt{3}}{3}$ , $-1$ , 0, 1.

Find the exact value of $\tan (30 \pi)$ using a coterminal angle. Options: -1, 0, 1, undefined.

Find $\sin \theta$ for the point $(4,-3)$ . Options: $-3$ , $\frac{4}{5}$ , $-\frac{3}{5}$ , $\frac{5}{4}$ .

Find $\csc B$ for a right triangle with sides $a=5$ and $b=6$ . Provide exact answers with rational denominators.

Find the exact value of $-\frac{\sec 10^{\circ}}{\csc 50^{\circ}}$ using identities. Choices: $-1$ , $1$ , $0$ , undefined.

Find the exact value of $\tan(-690^{\circ})$ using a coterminal angle, without a calculator. Choices: $\frac{\sqrt{3}}{3}$ , $\sqrt{3}$ , $-\sqrt{3}$ , $\frac{\sqrt{3}}{2}$ .

Given point $(-3,-4)$ , find $\sec \theta$ . Options: $-\frac{3}{5}$ , $-\frac{5}{3}$ , $\frac{5}{4}$ , $\frac{4}{3}$ .

Find the exact value of $\sin \frac{5 \pi}{3}$ using the reference angle, without a calculator. Options: $-\frac{1}{2}$ , $\frac{\sqrt{3}}{2}$ , $-1$ , $-\frac{\sqrt{3}}{2}$ .

Find $f(45^{\circ})$ for $f(\theta)=\sin \theta$ . What is the value? Options: $\frac{\sqrt{2}}{2}$ , $\frac{1}{2}$ , $\sqrt{2}$ , $-\frac{\sqrt{2}}{2}$ .

Find $\sec \theta$ for the point $P(-3,-1)$ on the circle $x^{2}+y^{2}=r^{2}$ .

Find $\theta$ in $[0^{\circ}, 90^{\circ}]$ such that $\tan \theta = 0.75248493$ . Calculate $\theta \approx$ .

Is the statement true or false? Evaluate if $1+\tan^{2} 30.1^{\circ} = -\sec^{2} 30.1^{\circ}$ .

Find the range of the tangent function: all reals except odd multiples of $\frac{\pi}{2}$ , or other options?

Determine the quadrant for angle $\theta$ where $\sin \theta > 0$ and $\cos \theta > 0$ .

If $\sin \theta=0.4$ , what is $\sin (\theta+\pi)$ ? Options: 0.4, -0.4, -0.6, 0.6.

Find the reference angle for the angle $-\frac{42 \pi}{8}$ .

Find $\tan \theta$ for the point $P(5,4)$ on the circle $x^{2}+y^{2}=r^{2}$ .

Find the exact value of $\sec \frac{3 \pi}{4}$ using the reference angle, without a calculator.

Find the reference angle for $-\frac{42 \pi}{8}$ . Options: $\frac{\pi}{4}$ , $\frac{\pi}{2}$ , $\frac{\pi}{3}$ , $\frac{\pi}{8}$ .

Find $\sin \theta$ for the point $P(-4,-3)$ on the circle $x^{2}+y^{2}=r^{2}$ . Choices: $-\frac{3}{5}$ , $\frac{3}{5}$ , $\frac{4}{5}$ , $-\frac{4}{5}$ .

Find $\cot \theta$ if $\cos \theta = \frac{21}{29}$ and $\frac{3 \pi}{2} < \theta < 2 \pi$ .

Find $\sin t$ for the point $P\left(-\frac{\sqrt{77}}{9},-\frac{2}{9}\right)$ on the unit circle.

Find $\tan t$ for the point $\mathrm{P} = \left(\frac{3}{8}, \frac{\sqrt{55}}{8}\right)$ on the unit circle.

Determine the quadrant for angle $\theta$ given that $\cos \theta > 0$ and $\csc \theta < 0$ .

Determine the quadrant for angle $\theta$ if $\csc \theta>0$ and $\sec \theta>0$ .

What is the range of the secant function? Options: 1) [-1, 1] 2) all reals except odd multiples of $\frac{\pi}{2}$ 3) $\geq 1$ or $\leq -1$ 4) all reals

Find the exact value of $\tan \frac{-3 \pi}{4}$ using the reference angle without a calculator.

Find the exact value of $\cot \frac{-11 \pi}{6}$ using the reference angle without a calculator.

Determine the quadrant for angle $\theta$ given that $\cot \theta<0$ and $\cos \theta>0$ .

What is the range of the sine function: $[-1, 1]$ ?

Determine the quadrant for angle $\theta$ if $\sec \theta<0$ and $\tan \theta<0$ .

Find the values of $\theta$ for which the function $f(\theta)=\sec \theta$ is undefined.

Find the values of $\theta$ for which $f(\theta)=\sec \theta$ is undefined.

Determine the quadrant for angle $\theta$ given that $\sin \theta > 0$ and $\cos \theta < 0$ .

Determine the quadrant for angle $\theta$ given that $\tan \theta>0$ and $\sin \theta<0$ .

Find $\sec \theta$ if $\sin \theta = -\frac{4}{9}$ and $\tan \theta > 0$ .

Find $\cos \theta$ and $\tan \theta$ given $\sin \theta=\frac{1}{2}$ and $\sec \theta<0$ .

Find the reference angle for $-\frac{5 \pi}{6}$ . Options: $\frac{\pi}{12}$ , $\frac{\pi}{6}$ , $\frac{7 \pi}{6}$ , $\frac{5 \pi}{6}$ .

Determine the quadrant for angle $\theta$ given that $\cot \theta>0$ and $\sin \theta<0$ .

Find $\cos \theta$ if $\tan \theta = -\frac{10}{7}$ and $\theta$ is in quadrant II.

In a right triangle with hypotenuse 100km and opposite side 60km, find $\sin \theta = \frac{60}{100}$ .

Find the quadrant for angle $\theta$ where $\tan \theta<0$ and $\sin \theta<0$ .

Determine the quadrant for angle $\theta$ given that $\cos \theta < 0$ and $\csc \theta < 0$ .

Find $\tan \theta$ given $\sec \theta = \frac{3}{2}$ and $\theta$ is in quadrant IV.

Find $\csc \theta$ given that $\sin \theta = \frac{1}{3}$ .

Find the reference angle for the angle $\frac{7 \pi}{8}$ .

Find $\cos \theta$ for the point $P(12, 5)$ on the circle $x^{2}+y^{2}=r^{2}$ .

Find $\cot \theta$ for the point $P(-3,2)$ on the circle $x^{2}+y^{2}=r^{2}$ .

Find $\cos \theta$ if $\tan \theta=-\frac{15}{8}$ and $90^{\circ}<\theta<180^{\circ}$ .

Find the sine of angle $t$ for the point $P\left(\frac{5}{8}, \frac{\sqrt{39}}{8}\right)$ on the unit circle.

Find the cosine of angle $t$ for the point $P\left(-\frac{\sqrt{55}}{8}, \frac{3}{8}\right)$ on the unit circle.

Find the domain of the sine function. What values can $x$ take for $\sin(x)$ ?

Find the values of $\theta$ for which $f(\theta)=\csc \theta$ is undefined.

Find the values of $\theta$ for which $f(\theta)=\csc \theta$ is undefined.

Find $\csc \theta$ given $\cot \theta = -\frac{9}{4}$ and $\cos \theta < 0$ .

Identify which trigonometric values are negative: I. $\sin \left(-292^{\circ}\right)$ , II. $\tan \left(-193^{\circ}\right)$ , III. $\cos \left(-207^{\circ}\right)$ , IV. $\cot 222^{\circ}$ .

If $\tan \theta=a$ (where $a \neq 0$ ), determine $\cot \theta$ .

If $\tan \theta=a(a \neq 0)$ , what is $\cot \theta$ using reciprocal identities?

Find the reference angle for the angle $-\frac{13 \pi}{12}$ .

Find the reference angle for $\frac{4 \pi}{3}$ . Options: $\frac{2 \pi}{3}$ , $\frac{\pi}{6}$ , $\frac{\pi}{3}$ , $\frac{4 \pi}{3}$ .

Find $\csc \theta$ for the point $P(-3,-1)$ on the circle $x^{2}+y^{2}=r^{2}$ .