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