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QuestionFind the exact values of sin⁡θ\sin \theta and cos⁡θ\cos \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 II3. We need to find the exact values of the remaining trigonometric functions of θ\theta, i.e., sin⁡θ\sin \theta and cos⁡θ\cos \theta
4. We will rationalize denominators when applicable

STEP 2

In a right triangle, the tangent of an angle θ\theta is defined as 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 tangent is negative) and the adjacent side is4.

STEP 3

The hypotenuse of the right triangle can be found using the Pythagorean theorem a2+b2=c2a^2 + b^2 = c^2, where aa and bb are the lengths of the two sides of the triangle and cc is the length of the hypotenuse.

STEP 4

Substitute the values of aa and bb into the Pythagorean theorem to find the hypotenuse.
c=(−3)2+42c = \sqrt{(-3)^2 +4^2}

STEP 5

Calculate the value of the hypotenuse.
c=(−3)2+42=9+16=25=5c = \sqrt{(-3)^2 +4^2} = \sqrt{9 +16} = \sqrt{25} =5

STEP 6

The sine of an angle θ\theta in a right triangle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse. So, sin⁡θ=oppositehypotenuse\sin \theta = \frac{opposite}{hypotenuse}.

STEP 7

Substitute the values of the opposite side and the hypotenuse into the formula to find sin⁡θ\sin \theta.
sin⁡θ=−35\sin \theta = \frac{-3}{5}

STEP 8

The cosine of an angle θ\theta in a right triangle is defined as the ratio of the length of the side adjacent to the angle to the length of the hypotenuse. So, cos⁡θ=adjacenthypotenuse\cos \theta = \frac{adjacent}{hypotenuse}.

STEP 9

Substitute the values of the adjacent side and the hypotenuse into the formula to find cos⁡θ\cos \theta.
cos⁡θ=45\cos \theta = \frac{4}{5}

STEP 10

Since θ\theta is in quadrant II, the cosine of θ\theta should be negative (as cosine is negative in the second quadrant). Therefore, cos⁡θ=−45\cos \theta = -\frac{4}{5}.
So, the exact values of the remaining trigonometric functions of θ\theta are sin⁡θ=−35\sin \theta = -\frac{3}{5} and cos⁡θ=−45\cos \theta = -\frac{4}{5}.

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