Math  /  Trigonometry

QuestionThe point is on the terminal side of an angle in standard position. Find the exact values of the six trigonometric functions of the (8,15)sinθ=cosθ=tanθ=cscθ=secθ=cotθ=\begin{array}{l} (8,15) \\ \sin \theta=\square \\ \cos \theta=\square \\ \tan \theta=\square \\ \csc \theta=\square \\ \sec \theta=\square \\ \cot \theta=\square \end{array}

Studdy Solution

STEP 1

1. The point (8,15)(8, 15) lies on the terminal side of an angle θ\theta in standard position.
2. The six trigonometric functions to find are sine, cosine, tangent, cosecant, secant, and cotangent.

STEP 2

1. Calculate the hypotenuse using the Pythagorean theorem.
2. Find the sine, cosine, and tangent of θ\theta.
3. Find the cosecant, secant, and cotangent of θ\theta.

STEP 3

Calculate the hypotenuse r r using the Pythagorean theorem:
r=x2+y2 r = \sqrt{x^2 + y^2}
Substitute the given values:
r=82+152 r = \sqrt{8^2 + 15^2} r=64+225 r = \sqrt{64 + 225} r=289 r = \sqrt{289} r=17 r = 17

STEP 4

Find the sine, cosine, and tangent of θ\theta:
sinθ=yr=1517\sin \theta = \frac{y}{r} = \frac{15}{17}
cosθ=xr=817\cos \theta = \frac{x}{r} = \frac{8}{17}
tanθ=yx=158\tan \theta = \frac{y}{x} = \frac{15}{8}

STEP 5

Find the cosecant, secant, and cotangent of θ\theta:
cscθ=1sinθ=1715\csc \theta = \frac{1}{\sin \theta} = \frac{17}{15}
secθ=1cosθ=178\sec \theta = \frac{1}{\cos \theta} = \frac{17}{8}
cotθ=1tanθ=815\cot \theta = \frac{1}{\tan \theta} = \frac{8}{15}
The exact values of the six trigonometric functions are:
sinθ=1517\sin \theta = \frac{15}{17}
cosθ=817\cos \theta = \frac{8}{17}
tanθ=158\tan \theta = \frac{15}{8}
cscθ=1715\csc \theta = \frac{17}{15}
secθ=178\sec \theta = \frac{17}{8}
cotθ=815\cot \theta = \frac{8}{15}

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