Studdy AI Logo
Math

QuestionFind the values of the six trigonometric functions for the angle θ\theta with point (−6,7)(-6,7) on its terminal side.

Studdy Solution

STEP 1

Assumptions1. The point (-6,7) lies on the terminal side of an angle θ\theta in standard position. . We are asked to find the exact values of the six trigonometric functions of θ\theta: sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot).
3. The point (-6,7) corresponds to the coordinates (x,y) in the Cartesian plane.

STEP 2

In the Cartesian plane, the x-coordinate corresponds to the length of the adjacent side (cosine) and the y-coordinate corresponds to the length of the opposite side (sine) in a right triangle. The hypotenuse (r) can be found using the Pythagorean theorem.
r=x2+y2r = \sqrt{x^2 + y^2}

STEP 3

Now, plug in the given values for x and y to calculate r.
r=(−6)2+72r = \sqrt{(-6)^2 +7^2}

STEP 4

Calculate the hypotenuse r.
r=(−6)2+72=36+49=85r = \sqrt{(-6)^2 +7^2} = \sqrt{36 +49} = \sqrt{85}

STEP 5

Now that we have the lengths of the sides, we can find the six trigonometric functions. Let's start with sine (sin), cosine (cos), and tangent (tan).
sin θ=yr, cos θ=xr, tan θ=yxsin\, \theta = \frac{y}{r},\, cos\, \theta = \frac{x}{r},\, tan\, \theta = \frac{y}{x}

STEP 6

Plug in the values for x, y, and r to calculate sin θ\theta, cos θ\theta, and tan θ\theta.
sin θ=85, cos θ=−685, tan θ=−6sin\, \theta = \frac{}{\sqrt{85}},\, cos\, \theta = \frac{-6}{\sqrt{85}},\, tan\, \theta = \frac{}{-6}

STEP 7

The other three trigonometric functions are the reciprocals of the first three. So, cosecant (csc) is the reciprocal of sine, secant (sec) is the reciprocal of cosine, and cotangent (cot) is the reciprocal of tangent.
csc θ=1sin θ, sec θ=1cos θ, cot θ=1tan θcsc\, \theta = \frac{1}{sin\, \theta},\, sec\, \theta = \frac{1}{cos\, \theta},\, cot\, \theta = \frac{1}{tan\, \theta}

STEP 8

Plug in the values for sin θ\theta, cos θ\theta, and tan θ\theta to calculate csc θ\theta, sec θ\theta, and cot θ\theta.
csc θ=1785, sec θ=1−685, cot θ=17−6csc\, \theta = \frac{1}{\frac{7}{\sqrt{85}}},\, sec\, \theta = \frac{1}{\frac{-6}{\sqrt{85}}},\, cot\, \theta = \frac{1}{\frac{7}{-6}}

STEP 9

Calculate the values for csc θ\theta, sec θ\theta, and cot θ\theta.
csc θ=857, sec θ=−856, cot θ=−67csc\, \theta = \frac{\sqrt{85}}{7},\, sec\, \theta = \frac{-\sqrt{85}}{6},\, cot\, \theta = \frac{-6}{7}The exact values of the six trigonometric functions of θ\theta aresin θ=785, cos θ=−685, tan θ=7−6sin\, \theta = \frac{7}{\sqrt{85}},\, cos\, \theta = \frac{-6}{\sqrt{85}},\, tan\, \theta = \frac{7}{-6}csc θ=857, sec θ=−856, cot θ=−67csc\, \theta = \frac{\sqrt{85}}{7},\, sec\, \theta = \frac{-\sqrt{85}}{6},\, cot\, \theta = \frac{-6}{7}

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