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Period of a function

The period     P of a repeating function f is the number representing the interval such that f ( x + P ) = f ( x ) for any value of x .

The period of the cosine, sine, secant, and cosecant functions is 2 π .

The period of the tangent and cotangent functions is π .

Finding the values of trigonometric functions

Find the values of the six trigonometric functions of angle t based on [link] .

This is an image of a graph of circle with angle of t inscribed. Point of (1/2, negative square root of 3 over 2) is at intersection of terminal side of angle and edge of circle.
sin  t = y = 3 2 cos  t = x = 1 2 tan  t = sin  t cos  t = 3 2 1 2 = 3 sec  t = 1 cos  t = 1 1 2 = −2 csc  t = 1 sin  t = 1 3 2 = 2 3 3 cot  t = 1 tan  t = 1 3 = 3 3
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Find the values of the six trigonometric functions of angle t based on [link] .

This is an image of a graph of circle with angle of t inscribed. Point of (0, -1) is at intersection of terminal side of angle and edge of circle.

sin t = 1 , cos t = 0 , tan t = Undefined sec t = Undefined, csc t = 1 , cot t = 0

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Finding the value of trigonometric functions

If sin ( t ) = 3 2 and cos ( t ) = 1 2 , find sec ( t ) , csc ( t ) , tan ( t ) , cot ( t ) .

sec  t = 1 cos  t = 1 1 2 = 2 csc  t = 1 sin  t = 1 3 2 2 3 3 tan  t = sin  t cos  t = 3 2 1 2 = 3 cot  t = 1 tan  t = 1 3 = 3 3
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sin ( t ) = 2 2 and cos ( t ) = 2 2 , find sec ( t ) , csc ( t ) , tan ( t ) , and cot ( t )

sec t = 2 , csc t = 2 , tan t = 1 , cot t = 1

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Evaluating trigonometric functions with a calculator

We have learned how to evaluate the six trigonometric functions for the common first-quadrant angles and to use them as reference angles for angles in other quadrants. To evaluate trigonometric functions of other angles, we use a scientific or graphing calculator or computer software. If the calculator has a degree mode and a radian mode, confirm the correct mode is chosen before making a calculation.

Evaluating a tangent function with a scientific calculator as opposed to a graphing calculator or computer algebra system is like evaluating a sine or cosine: Enter the value and press the TAN key. For the reciprocal functions, there may not be any dedicated keys that say CSC, SEC, or COT. In that case, the function must be evaluated as the reciprocal of a sine, cosine, or tangent.

If we need to work with degrees and our calculator or software does not have a degree mode, we can enter the degrees multiplied by the conversion factor π 180 to convert the degrees to radians. To find the secant of 30° , we could press

(for a scientific calculator): 1 30 × π 180 COS or (for a graphing calculator): 1 cos ( 30 π 180 )

Given an angle measure in radians, use a scientific calculator to find the cosecant.

  1. If the calculator has degree mode and radian mode, set it to radian mode.
  2. Enter: 1 /
  3. Enter the value of the angle inside parentheses.
  4. Press the SIN key.
  5. Press the = key.

Given an angle measure in radians, use a graphing utility/calculator to find the cosecant.

  • If the graphing utility has degree mode and radian mode, set it to radian mode.
  • Enter: 1 /
  • Press the SIN key.
  • Enter the value of the angle inside parentheses.
  • Press the ENTER key.

Evaluating the cosecant using technology

Evaluate the cosecant of 5 π 7 .

For a scientific calculator, enter information as follows:

1/ ( 5 × π / 7 )  SIN = csc ( 5 π 7 ) 1.279
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Evaluate the cotangent of π 8 .

2.414

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Access these online resources for additional instruction and practice with other trigonometric functions.

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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