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QuestionFind the remaining trigonometric functions of θ\theta if tan⁡θ=−34\tan \theta=-\frac{3}{4} in quadrant II.

Studdy Solution

STEP 1

Assumptions1. The given value of tan⁡θ\tan \theta is −34-\frac{3}{4}. . θ\theta is in quadrant II.
3. We need to find the exact value of the remaining trigonometric functions of θ\theta.

STEP 2

In any right triangle, the tangent of an angle is the ratio of the opposite side to the adjacent side. Given that tan⁡θ=−4\tan \theta = -\frac{}{4}, we can consider a right triangle where the opposite side is - (negative because θ\theta is in quadrant II where the tangent is negative) and the adjacent side is4.

STEP 3

We can use the Pythagorean theorem to find the length of the hypotenuse. The Pythagorean theorem states that in any right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
Hypotenuse=Opposite2+Adjacent2Hypotenuse = \sqrt{Opposite^2 + Adjacent^2}

STEP 4

Plug in the values for the opposite and adjacent sides to calculate the hypotenuse.
Hypotenuse=(−3)2+42Hypotenuse = \sqrt{(-3)^2 +4^2}

STEP 5

Calculate the length of the hypotenuse.
Hypotenuse=9+16=25=5Hypotenuse = \sqrt{9 +16} = \sqrt{25} =5

STEP 6

Now that we have the lengths of all sides of the triangle, we can find the exact values of the remaining trigonometric functions. The sine of an angle in a right triangle is the ratio of the length of the opposite side to the length of the hypotenuse.
sin⁡θ=OppositeHypotenuse\sin \theta = \frac{Opposite}{Hypotenuse}

STEP 7

Plug in the values for the opposite side and the hypotenuse to calculate sin⁡θ\sin \theta.
sin⁡θ=−35\sin \theta = \frac{-3}{5}

STEP 8

The cosine of an angle in a right triangle is the ratio of the length of the adjacent side to the length of the hypotenuse.
cos⁡θ=AdjacentHypotenuse\cos \theta = \frac{Adjacent}{Hypotenuse}

STEP 9

Plug in the values for the adjacent side and the hypotenuse to calculate cos⁡θ\cos \theta.
cos⁡θ=45\cos \theta = \frac{4}{5}

STEP 10

The cosecant, secant, and cotangent are the reciprocals of the sine, cosine, and tangent, respectively.
csc⁡θ=sin⁡θ\csc \theta = \frac{}{\sin \theta}sec⁡θ=cos⁡θ\sec \theta = \frac{}{\cos \theta}cot⁡θ=tan⁡θ\cot \theta = \frac{}{\tan \theta}

STEP 11

Plug in the values for sin⁡θ\sin \theta, cos⁡θ\cos \theta, and tan⁡θ\tan \theta to calculate csc⁡θ\csc \theta, sec⁡θ\sec \theta, and cot⁡θ\cot \theta.
csc⁡θ=−3/5=−53\csc \theta = \frac{}{-3/5} = -\frac{5}{3}sec⁡θ=4/5=54\sec \theta = \frac{}{4/5} = \frac{5}{4}cot⁡θ=−3/4=−43\cot \theta = \frac{}{-3/4} = -\frac{4}{3}So, the exact values of the trigonometric functions are sin⁡θ=−35\sin \theta = -\frac{3}{5}, cos⁡θ=45\cos \theta = \frac{4}{5}, csc⁡θ=−53\csc \theta = -\frac{5}{3}, sec⁡θ=54\sec \theta = \frac{5}{4}, and cot⁡θ=−43\cot \theta = -\frac{4}{3}.

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