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Math  /  Trigonometry

QuestionFind sec⁡θ\sec \theta and csc⁡θ\csc \theta if cot⁡θ=−1235\cot \theta=-\frac{12}{35} and cos⁡θ<0\cos \theta<0.

Studdy Solution

STEP 1

1. The trigonometric identity cot⁡θ=cos⁡θsin⁡θ\cot \theta = \frac{\cos \theta}{\sin \theta} will be used.
2. The Pythagorean identity sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1 will help find sin⁡θ\sin \theta and cos⁡θ\cos \theta.
3. The quadrant of θ\theta will be determined using cos⁡θ<0\cos \theta < 0 and cot⁡θ<0\cot \theta < 0.

STEP 2

1. Determine the quadrant where θ\theta lies.
2. Express sin⁡θ\sin \theta and cos⁡θ\cos \theta using the given cot⁡θ\cot \theta.
3. Use the Pythagorean identity to find sin⁡θ\sin \theta and cos⁡θ\cos \theta.
4. Calculate sec⁡θ\sec \theta and csc⁡θ\csc \theta.

STEP 3

Determine the quadrant where θ\theta lies. Given cot⁡θ=−1235\cot \theta = -\frac{12}{35} and cos⁡θ<0\cos \theta < 0, we know: - cot⁡θ<0\cot \theta < 0 implies θ\theta is in either the second or fourth quadrant. - cos⁡θ<0\cos \theta < 0 implies θ\theta is in the second or third quadrant.
Thus, θ\theta must be in the second quadrant.

STEP 4

Express sin⁡θ\sin \theta and cos⁡θ\cos \theta using cot⁡θ=−1235\cot \theta = -\frac{12}{35}.
Since cot⁡θ=cos⁡θsin⁡θ=−1235\cot \theta = \frac{\cos \theta}{\sin \theta} = -\frac{12}{35}, we can set: cos⁡θ=−12kandsin⁡θ=35k\cos \theta = -12k \quad \text{and} \quad \sin \theta = 35k for some positive kk (since sin⁡θ>0\sin \theta > 0 in the second quadrant).

STEP 5

Use the Pythagorean identity to find kk.
Substitute cos⁡θ\cos \theta and sin⁡θ\sin \theta into the identity: (−12k)2+(35k)2=1(-12k)^2 + (35k)^2 = 1 144k2+1225k2=1144k^2 + 1225k^2 = 1 1369k2=11369k^2 = 1 k2=11369k^2 = \frac{1}{1369} k=137k = \frac{1}{37}
Now find sin⁡θ\sin \theta and cos⁡θ\cos \theta: sin⁡θ=35k=3537\sin \theta = 35k = \frac{35}{37} cos⁡θ=−12k=−1237\cos \theta = -12k = -\frac{12}{37}

STEP 6

Calculate sec⁡θ\sec \theta and csc⁡θ\csc \theta.
sec⁡θ=1cos⁡θ=−3712\sec \theta = \frac{1}{\cos \theta} = -\frac{37}{12} csc⁡θ=1sin⁡θ=3735\csc \theta = \frac{1}{\sin \theta} = \frac{37}{35}
The values are: sec⁡θ=−3712,csc⁡θ=3735\sec \theta = -\frac{37}{12}, \quad \csc \theta = \frac{37}{35}

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