The point #(8,15)# is on the terminal side of an angle in standard position, how do you determine the exact values of the six trigonometric functions of the angle?

Answer 1

#sin theta = 15/17; cos theta = 8/17#
#tan theta = 15/8; cot theta = 8/15#
#csc theta = 17/15; sec theta = 17/8#

Draw a right triangle in the first quadrant of the rectangular coordinate plane with base = #8# and height = #15#.
Calculate the hypotenuse using Pythagorean Theorem: #r = sqrt (8^2 + 15^2) = sqrt(64 + 225+ = sqrt(289) = 17#
Use the trig. definitions to find all of the angles: #sin theta = "opposite"/"hypotenuse" = 15/17#
#cos theta = "adjacent"/"hypotenuse" = 8/17#
#tan theta = "opposite"/"adjacent" = 15/8#
#csc theta = 1/(sin theta) = 17/15#
#sec theta = 1/(cos theta) = 17/8#
#cot theta = 1/(tan theta) = 8/15#
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Answer 2

To determine the exact values of the six trigonometric functions of the angle corresponding to the point (8,15) on the terminal side in standard position, we can use the following steps:

  1. First, calculate the length of the radius (r) using the Pythagorean theorem: ( r = \sqrt{x^2 + y^2} = \sqrt{8^2 + 15^2} = \sqrt{64 + 225} = \sqrt{289} = 17 )

  2. Next, determine the values of the trigonometric functions using the coordinates (x, y) and the radius (r):

    • ( \sin(\theta) = \frac{y}{r} = \frac{15}{17} )
    • ( \cos(\theta) = \frac{x}{r} = \frac{8}{17} )
    • ( \tan(\theta) = \frac{y}{x} = \frac{15}{8} )
    • ( \csc(\theta) = \frac{1}{\sin(\theta)} = \frac{17}{15} )
    • ( \sec(\theta) = \frac{1}{\cos(\theta)} = \frac{17}{8} )
    • ( \cot(\theta) = \frac{1}{\tan(\theta)} = \frac{8}{15} )

Therefore, the exact values of the six trigonometric functions of the angle corresponding to the point (8,15) on the terminal side in standard position are:

  • ( \sin(\theta) = \frac{15}{17} )
  • ( \cos(\theta) = \frac{8}{17} )
  • ( \tan(\theta) = \frac{15}{8} )
  • ( \csc(\theta) = \frac{17}{15} )
  • ( \sec(\theta) = \frac{17}{8} )
  • ( \cot(\theta) = \frac{8}{15} )
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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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