Math  /  Trigonometry

QuestionMultiple Choice: Circle the correct answer. 1) The exact value of cos2π12+sin2π12\cos ^{2} \frac{\pi}{12}+\sin ^{2} \frac{\pi}{12} is... a) 12\frac{1}{2} b) 62\frac{\sqrt{6}}{2} c) 1 d) 32\frac{3}{2} 2) The expression sin(π2x)\sin \left(\frac{\pi}{2}-x\right) is equivalent to... a) sinxsinπ2\sin x-\sin \frac{\pi}{2} b) cosx\cos x c) cos(xπ2)\cos \left(x-\frac{\pi}{2}\right) d) cscx\csc x 3) The exact degree measure for the angle 5π18\frac{5 \pi}{18} is... a) 5π18\frac{5 \pi^{\circ}}{18} b) 5050^{\circ} c) 518\frac{5}{18} 。 d) 900900^{\circ} 4) The exact value of sin7π4\sin \frac{7 \pi}{4} is... a) 12-\frac{1}{\sqrt{2}} b) 1 c) -1 d) 12\frac{1}{\sqrt{2}} 5) The exact value of cos7π6\cos \frac{7 \pi}{6} is... a) 12-\frac{1}{2} b) 12\frac{1}{2} c) 32\frac{\sqrt{3}}{2} d) 32-\frac{\sqrt{3}}{2} 6) The exact value of cotπ2\cot \frac{\pi}{2} is... a) 0 b) -1 c) 1 d) undefin

Studdy Solution

STEP 1

1. We are using standard trigonometric identities and angle conversions.
2. We are working with radians and degrees where applicable.

STEP 2

1. Solve for the exact value of cos2π12+sin2π12\cos^2 \frac{\pi}{12} + \sin^2 \frac{\pi}{12}.
2. Determine the equivalent expression for sin(π2x)\sin \left(\frac{\pi}{2} - x\right).
3. Convert 5π18\frac{5\pi}{18} radians to degrees.
4. Find the exact value of sin7π4\sin \frac{7\pi}{4}.
5. Find the exact value of cos7π6\cos \frac{7\pi}{6}.
6. Determine the exact value of cotπ2\cot \frac{\pi}{2}.

STEP 3

Use the Pythagorean identity: cos2θ+sin2θ=1\cos^2 \theta + \sin^2 \theta = 1 for any angle θ\theta.
For θ=π12\theta = \frac{\pi}{12}:
cos2π12+sin2π12=1\cos^2 \frac{\pi}{12} + \sin^2 \frac{\pi}{12} = 1
The correct answer is c) 1.

STEP 4

Use the co-function identity: sin(π2x)=cosx\sin \left(\frac{\pi}{2} - x\right) = \cos x.
The correct answer is b) cosx\cos x.

STEP 5

Convert radians to degrees using the formula: degrees=radians×180π \text{degrees} = \text{radians} \times \frac{180}{\pi} .
For 5π18\frac{5\pi}{18}:
5π18×180π=50\frac{5\pi}{18} \times \frac{180}{\pi} = 50^\circ
The correct answer is b) 5050^\circ.

STEP 6

Determine the sine of 7π4\frac{7\pi}{4}. Recognize that 7π4\frac{7\pi}{4} is in the fourth quadrant where sine is negative, and it is equivalent to π4-\frac{\pi}{4}.
sin7π4=sinπ4=12\sin \frac{7\pi}{4} = -\sin \frac{\pi}{4} = -\frac{1}{\sqrt{2}}
The correct answer is a) 12-\frac{1}{\sqrt{2}}.

STEP 7

Determine the cosine of 7π6\frac{7\pi}{6}. Recognize that 7π6\frac{7\pi}{6} is in the third quadrant where cosine is negative.
cos7π6=cosπ6=32\cos \frac{7\pi}{6} = -\cos \frac{\pi}{6} = -\frac{\sqrt{3}}{2}
The correct answer is d) 32-\frac{\sqrt{3}}{2}.

STEP 8

Determine the cotangent of π2\frac{\pi}{2}. Recognize that cotθ=cosθsinθ\cot \theta = \frac{\cos \theta}{\sin \theta} and sinπ2=1\sin \frac{\pi}{2} = 1, cosπ2=0\cos \frac{\pi}{2} = 0.
cotπ2=cosπ2sinπ2=01=undefined\cot \frac{\pi}{2} = \frac{\cos \frac{\pi}{2}}{\sin \frac{\pi}{2}} = \frac{0}{1} = \text{undefined}
The correct answer is d) \text{undefin}.

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