Math

QuestionFind the remaining trigonometric functions of θ\theta if sinθ=37\sin \theta=-\frac{\sqrt{3}}{7} in quadrant III.

Studdy Solution

STEP 1

Assumptions1. The given value of sinθ\sin \theta is 37-\frac{\sqrt{3}}{7} . θ\theta is in quadrant III3. We are looking for the exact values of the remaining trigonometric functions of θ\theta.

STEP 2

In the third quadrant, both sine and cosine are negative, and tangent is positive. Since we know the value of sine, we can use the Pythagorean identity to find the value of cosine. The Pythagorean identity issin2θ+cos2θ=1\sin^2 \theta + \cos^2 \theta =1

STEP 3

Substitute the given value of sinθ\sin \theta into the Pythagorean identity.
(37)2+cos2θ=1\left(-\frac{\sqrt{3}}{7}\right)^2 + \cos^2 \theta =1

STEP 4

implify the equation.
349+cos2θ=1\frac{3}{49} + \cos^2 \theta =1

STEP 5

Rearrange the equation to solve for cos2θ\cos^2 \theta.
cos2θ=1349\cos^2 \theta =1 - \frac{3}{49}

STEP 6

implify the right side of the equation.
cos2θ=4649\cos^2 \theta = \frac{46}{49}

STEP 7

Take the square root of both sides to solve for cosθ\cos \theta. Remember that in the third quadrant, cosine is negative.
cosθ=467\cos \theta = -\frac{\sqrt{46}}{7}So, the correct choice is A cosθ=467\cos \theta = -\frac{\sqrt{46}}{7}

STEP 8

Next, we find the value of cscθ\csc \theta. The cosecant function is the reciprocal of the sine function.
cscθ=1sinθ\csc \theta = \frac{1}{\sin \theta}

STEP 9

Substitute the given value of sinθ\sin \theta into the equation.
cscθ=37\csc \theta = \frac{}{-\frac{\sqrt{3}}{7}}

STEP 10

implify the equation to find the value of cscθ\csc \theta.
cscθ=73\csc \theta = -\frac{7}{\sqrt{3}}

STEP 11

Rationalize the denominator by multiplying the numerator and denominator by 3\sqrt{3}.
cscθ=733\csc \theta = -\frac{7\sqrt{3}}{3}So, the correct choice is A cscθ=733\csc \theta = -\frac{7\sqrt{3}}{3}

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