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Angle θ is in standard position. If (8, -15) is on the terminal ray of angle θ, find the values of the trigonometric functions.

Angle θ is in standard position. If (8, -15) is on the terminal ray of angle θ, find-example-1
User She
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1 Answer

1 vote

Answer:

* sin Ф = -15/17 * cos Ф = 8/17 * tan Ф = -15/8

* csc Ф = -17/15 * sec Ф = 17/8 * cot Ф = -8/15

Explanation:

Lets revise the trigonometric function of angle Ф

- Angle θ is in standard position

- Point (8, -15) is on the terminal ray of angle θ

- That means the terminal is the hypotenuse of a right triangle x and y

are its legs

∵ x-coordinate is positive and y-coordinate is negative

∴ angle Ф lies in the 4th quadrant

- The opposite side of angle Ф is the y-coordinate of the point on the

terminal ray of angle Ф and the adjacent side to angle Ф is the

x-coordinate of that point

∵ The length of the hypotenuse (h) = √(x² + y²)

∴ h = √[(8)² + (-15)²] = √[64 + 225] = √[289] = 17

∴ The length of the hypotenuse is 17

- Lets find sin Ф

∵ sin Ф = opposite/hypotenuse

∵ The opposite is y = -15

∵ The hypotenuse = 17

sin Ф = -15/17

- Lets find cos Ф

∵ cos Ф = adjacent/hypotenuse

∵ The adjacent is x = 8

∵ The hypotenuse = 17

cos Ф = 8/17

- Lets find tan Ф

∵ tan Ф = opposite/adjacent

∵ The opposite is y = -15

∵ The adjacent = 8

tan Ф = -15/8

- Remember csc Ф is the reciprocal of sin Ф

∵ csc Ф = 1/sin Ф

∵ sin Ф = -15/17

csc Ф = -17/15

- Remember sec Ф is the reciprocal of cos Ф

∵ sec Ф = 1/ cos Ф

∵ cos Ф = 8/17

sec Ф = 17/8

- Remember cot Ф is the reciprocal of tan Ф

∵ cot Ф = 1/tan Ф

∵ tan Ф = -15/8

∴ cot Ф = -8/15

User Ickydime
by
8.2k points

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