Studdy AI Logo
Math

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

Studdy Solution

STEP 1

Assumptions1. The given value of sin⁡θ\sin \theta is −37-\frac{\sqrt{3}}{7} . θ\theta is in quadrant III3. We are to find the exact values of the remaining trigonometric functions of θ\theta, which are cos⁡θ\cos \theta, tan⁡θ\tan \theta, csc⁡θ\csc \theta, sec⁡θ\sec \theta, and cot⁡θ\cot \theta.

STEP 2

In any right triangle, the sine of an angle is defined as the length of the opposite side divided by the length of the hypotenuse. Since we know that sin⁡θ=−7\sin \theta = -\frac{\sqrt{}}{7}, we can consider a right triangle where the opposite side is −-\sqrt{} (the negative sign indicates direction, not length) and the hypotenuse is 77.

STEP 3

We can use the Pythagorean theorem to find the length of the adjacent side. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. We can write this asAdjacent2=Hypotenuse2−Opposite2Adjacent^2 = Hypotenuse^2 - Opposite^2

STEP 4

Plug in the values for the hypotenuse and the opposite side to calculate the square of the length of the adjacent side.
Adjacent2=72−(−3)2Adjacent^2 =7^2 - (-\sqrt{3})^2

STEP 5

Calculate the square of the length of the adjacent side.
Adjacent2=49−3=46Adjacent^2 =49 -3 =46

STEP 6

Take the square root of both sides to find the length of the adjacent side. Note that since θ\theta is in quadrant III, the adjacent side is negative.
Adjacent=−46Adjacent = -\sqrt{46}

STEP 7

Now that we have the lengths of all sides of the triangle, we can find the values of the remaining trigonometric functions. The cosine of an angle is defined as the length of the adjacent side divided by the length of the hypotenuse, so we havecos⁡θ=AdjacentHypotenuse\cos \theta = \frac{Adjacent}{Hypotenuse}

STEP 8

Plug in the values for the adjacent side and the hypotenuse to calculate cos⁡θ\cos \theta.
cos⁡θ=−467\cos \theta = \frac{-\sqrt{46}}{7}

STEP 9

The tangent of an angle is defined as the sine of the angle divided by the cosine of the angle, so we havetan⁡θ=sin⁡θcos⁡θ\tan \theta = \frac{\sin \theta}{\cos \theta}

STEP 10

Plug in the values for sin⁡θ\sin \theta and cos⁡θ\cos \theta to calculate tan⁡θ\tan \theta.
tan⁡θ=−3/7−46/7\tan \theta = \frac{-\sqrt{3}/7}{-\sqrt{46}/7}

STEP 11

implify the expression to find the value of tan⁡θ\tan \theta.
tan⁡θ=346=13846\tan \theta = \frac{\sqrt{3}}{\sqrt{46}} = \frac{\sqrt{138}}{46}

STEP 12

The cotangent of an angle is defined as the reciprocal of the tangent of the angle, so we havecot⁡θ=tan⁡θ\cot \theta = \frac{}{\tan \theta}

STEP 13

Plug in the value for tan⁡θ\tan \theta to calculate cot⁡θ\cot \theta.
cot⁡θ=138/46\cot \theta = \frac{}{\sqrt{138}/46}

STEP 14

implify the expression to find the value of cot⁡θ\cot \theta.
cot⁡θ=46138=46138138\cot \theta = \frac{46}{\sqrt{138}} = \frac{46\sqrt{138}}{138}So, the exact values of the remaining trigonometric functions of θ\theta arecos⁡θ=−467\cos \theta = -\frac{\sqrt{46}}{7}, tan⁡θ=13846\tan \theta = \frac{\sqrt{138}}{46}, cot⁡θ=46138138\cot \theta = \frac{46\sqrt{138}}{138}.

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord