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Solved on Feb 11, 2024

Find the exact values of sin⁡θ\sin \theta and csc⁡θ\csc \theta given tan⁡θ=8/15\tan \theta = 8/15 and cos⁡θ<0\cos \theta < 0.

STEP 1

Assumptions
1. We are given that tan⁡θ=815\tan \theta = \frac{8}{15}.
2. We are also given that cos⁡θ<0\cos \theta < 0.
3. We need to find the exact values of the remaining trigonometric functions of θ\theta: sin⁡θ\sin \theta, cos⁡θ\cos \theta, csc⁡θ\csc \theta, sec⁡θ\sec \theta, and cot⁡θ\cot \theta.
4. We assume that θ\theta is in the second quadrant because tan⁡θ\tan \theta is positive and cos⁡θ\cos \theta is negative in that quadrant.

STEP 2

Recall the identity that relates tan⁡θ\tan \theta to sin⁡θ\sin \theta and cos⁡θ\cos \theta:
tan⁡θ=sin⁡θcos⁡θ\tan \theta = \frac{\sin \theta}{\cos \theta}

STEP 3

Since we know tan⁡θ=815\tan \theta = \frac{8}{15}, we can write:
sin⁡θcos⁡θ=815\frac{\sin \theta}{\cos \theta} = \frac{8}{15}

STEP 4

We will use the Pythagorean identity to find sin⁡θ\sin \theta and cos⁡θ\cos \theta:
sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1

STEP 5

Let x=cos⁡θx = \cos \theta and y=sin⁡θy = \sin \theta. Then, we can write the Pythagorean identity as:
y2+x2=1y^2 + x^2 = 1

STEP 6

From STEP_3, we have y=815xy = \frac{8}{15}x. Substitute this into the Pythagorean identity:
(815x)2+x2=1\left(\frac{8}{15}x\right)^2 + x^2 = 1

STEP 7

Simplify the equation:
64225x2+x2=1\frac{64}{225}x^2 + x^2 = 1

STEP 8

Combine like terms:
64225x2+225225x2=1\frac{64}{225}x^2 + \frac{225}{225}x^2 = 1

STEP 9

Add the fractions:
289225x2=1\frac{289}{225}x^2 = 1

STEP 10

Solve for x2x^2:
x2=225289x^2 = \frac{225}{289}

STEP 11

Since cos⁡θ<0\cos \theta < 0, we take the negative square root for xx:
x=cos⁡θ=−225289x = \cos \theta = -\sqrt{\frac{225}{289}}

STEP 12

Simplify the square root:
cos⁡θ=−1517\cos \theta = -\frac{15}{17}

STEP 13

Now, use the relation from STEP_3 to find y=sin⁡θy = \sin \theta:
y=815xy = \frac{8}{15}x

STEP 14

Substitute the value of xx:
sin⁡θ=815⋅−1517\sin \theta = \frac{8}{15} \cdot -\frac{15}{17}

STEP 15

Simplify the expression:
sin⁡θ=−817\sin \theta = -\frac{8}{17}

STEP 16

Next, we find the values of the other trigonometric functions. Recall the definitions of csc⁡θ\csc \theta, sec⁡θ\sec \theta, and cot⁡θ\cot \theta:
csc⁡θ=1sin⁡θ,sec⁡θ=1cos⁡θ,cot⁡θ=1tan⁡θ\csc \theta = \frac{1}{\sin \theta}, \quad \sec \theta = \frac{1}{\cos \theta}, \quad \cot \theta = \frac{1}{\tan \theta}

STEP 17

Calculate csc⁡θ\csc \theta:
csc⁡θ=1sin⁡θ=1−817\csc \theta = \frac{1}{\sin \theta} = \frac{1}{-\frac{8}{17}}

STEP 18

Simplify the expression:
csc⁡θ=−178\csc \theta = -\frac{17}{8}

STEP 19

Calculate sec⁡θ\sec \theta:
sec⁡θ=1cos⁡θ=1−1517\sec \theta = \frac{1}{\cos \theta} = \frac{1}{-\frac{15}{17}}

STEP 20

Simplify the expression:
sec⁡θ=−1715\sec \theta = -\frac{17}{15}

STEP 21

Calculate cot⁡θ\cot \theta:
cot⁡θ=1tan⁡θ=1815\cot \theta = \frac{1}{\tan \theta} = \frac{1}{\frac{8}{15}}

STEP 22

Simplify the expression:
cot⁡θ=158\cot \theta = \frac{15}{8}
The exact values of the remaining trigonometric functions of θ\theta are:
sin⁡θ=−817,cos⁡θ=−1517,csc⁡θ=−178,sec⁡θ=−1715,cot⁡θ=158\sin \theta = -\frac{8}{17}, \quad \cos \theta = -\frac{15}{17}, \quad \csc \theta = -\frac{17}{8}, \quad \sec \theta = -\frac{17}{15}, \quad \cot \theta = \frac{15}{8}

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