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QuestionFind all trigonometric functions for 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 the second quadrant.
3. We need to find the exact values of the remaining trigonometric functions of θ\theta.
4. We need to rationalize denominators when applicable.

STEP 2

We can use the Pythagorean identity 1+tan⁡2θ=sec⁡2θ1 + \tan^2 \theta = \sec^2 \theta to find the value of sec⁡θ\sec \theta.

STEP 3

Substitute the given value of tan⁡θ\tan \theta into the Pythagorean identity.
1+(−3)2=sec⁡2θ1 + \left(-\frac{3}{}\right)^2 = \sec^2 \theta

STEP 4

Calculate the value on the left side of the equation.
1+(−34)2=1+916=2516=sec⁡2θ1 + \left(-\frac{3}{4}\right)^2 =1 + \frac{9}{16} = \frac{25}{16} = \sec^2 \theta

STEP 5

Take the square root of both sides to find sec⁡θ\sec \theta. Remember that since θ\theta is in the second quadrant, sec⁡θ\sec \theta should be negative.
sec⁡θ=−2516=−54\sec \theta = -\sqrt{\frac{25}{16}} = -\frac{5}{4}

STEP 6

Now, we can find the value of csc⁡θ\csc \theta using the identity csc⁡θ=1sin⁡θ\csc \theta = \frac{1}{\sin \theta} and the fact that sin⁡θ=1sec⁡θ\sin \theta = \frac{1}{\sec \theta}.

STEP 7

Substitute the value of sec⁡θ\sec \theta into the equation for sin⁡θ\sin \theta.
sin⁡θ=1sec⁡θ=1−54=−45\sin \theta = \frac{1}{\sec \theta} = \frac{1}{-\frac{5}{4}} = -\frac{4}{5}

STEP 8

Substitute the value of sin⁡θ\sin \theta into the equation for csc⁡θ\csc \theta.
csc⁡θ=1sin⁡θ=1−45=−54\csc \theta = \frac{1}{\sin \theta} = \frac{1}{-\frac{4}{5}} = -\frac{5}{4}

STEP 9

Finally, we can find the value of cot⁡θ\cot \theta using the identity cot⁡θ=tan⁡θ\cot \theta = \frac{}{\tan \theta}.

STEP 10

Substitute the given value of tan⁡θ\tan \theta into the equation for cot⁡θ\cot \theta.
cot⁡θ=tan⁡θ=−34=−43\cot \theta = \frac{}{\tan \theta} = \frac{}{-\frac{3}{4}} = -\frac{4}{3}So, the exact values of the remaining trigonometric functions of θ\theta are sec⁡θ=−54\sec \theta = -\frac{5}{4}, csc⁡θ=−54\csc \theta = -\frac{5}{4}, and cot⁡θ=−43\cot \theta = -\frac{4}{3}.

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