1. Reciprocal Trigonometric Functions

2. Reciprocal The Reciprocal of any number x is: Ex: Find the reciprocal of 0.5x 2 Equivalent forms of this definition: The inverse of x is x -1. OR x and y are inverses if xy=1 Because:

3. Cosecant The cosecant function is the reciprocal of sine. It is defined as the following: Ex: Find the exact answer of csc(45°) There is no cosecant button on the calculator. You must use the definition above. CSC is the Abbreviation for cosecant

4. The Sine Function Graph In order to investigate the cosecant function, first examine the sine graph. Now find and graph all of the reciprocal values of sine. XSIN(X) -2π 0 -11π/6 0.5 -3π/2 1 -7π/6 0.5 -π-π 0 -5π/6 -0.5 -π/2 -π/6 -0.5 00 π/6 0.5 π/2 1 5π/6 0.5 π 0 7π/6 -0.5 3π/2 11π/6 -0.5 2π2π 0

5. The Cosecant Function Graph Find the reciprocal of the sine values. XSIN(X) -2π 0 -11π/6 0.5 -3π/2 1 -7π/6 0.5 -π-π 0 -5π/6 -0.5 -π/2 -π/6 -0.5 00 π/6 0.5 π/2 1 5π/6 0.5 π 0 7π/6 -0.5 3π/2 11π/6 -0.5 2π2π 0 CSC(X) 1/0 = DNE 1/0.5 = 2 1/1 = 1 1/0.5 = 2 1/0 = DNE 1/-0.5 = -2 1/-1 = -1 1/-0.5 = -2 1/0 = DNE 1/0.5 = 2 1/1 = 1 1/0.5 = 2 1/0 = DNE 1/-0.5 = -2 1/-1 = -1 1/-0.5 = -2 1/0 = DNE Plot the points.The errors are asymptotes.

6. The Cosecant Function Graph Domain: Range: Asymptotes All Reals except multiples of Pi. X = all multiples of Pi.

7. Secant The secant function is the reciprocal of cosine. It is defined as the following: Ex: Evaluate sec(72°) There is no secant button on the calculator. You must use the definition above. SEC is the Abbreviation for secant

8. The Cosine Function Graph In order to investigate the secant function, first examine the cosine graph. Now find and graph all of the reciprocal values of cosine. XCOS(X) -2π 1 -5π/3 0.5 -3π/2 0 -4π/3 -0.5 -π-π -2π/3 -0.5 -π/2 0 -π/3 0.5 01 π/3 0.5 π/2 0 2π/3 -0.5 π 4π/3 -0.5 3π/2 0 5π/3 0.5 2π2π 1

9. The Secant Function Graph Find the reciprocal of the cosine values. XCOS(X) -2π 1 -5π/3 0.5 -3π/2 0 -4π/3 -0.5 -π-π -2π/3 -0.5 -π/2 0 -π/3 0.5 01 π/3 0.5 π/2 0 2π/3 -0.5 π 4π/3 -0.5 3π/2 0 5π/3 0.5 2π2π 1 SEC(X) 1/1 = 1 1/0.5 = 2 1/0 = DNE 1/-0.5 = -2 1/-1 = -1 1/-0.5 = -2 1/0 = DNE 1/0.5 = 2 1/1 = 1 1/0.5 = 2 1/0 = DNE 1/-0.5 = -2 1/-1 = -1 1/-0.5 = -2 1/0 = DNE 1/0.5 = 2 1/1 = 1 Plot the points.The errors are asymptotes.

10. The Secant Function Graph Domain: Range: Asymptotes All Reals except

11. Cotangent The cotangent function is the reciprocal of tangent. It is defined as the following: Ex: Evaluate cot(0.5π) There is no secant button on the calculator. You must use the definition above. COT is the Abbreviation for secant The best form to use.

12. The Tangent Function Graph In order to investigate the cotangent function, first examine the tangent graph and all the values of cosine and sine. Remember, cotangent is cosine divided by sine. Now find and graph all of the values of cosine÷sine. XCOS(X)SIN(X) -2π 10 -7π/4 0.707 -3π/2 01 -5π/4 -0.7070.707 -π-π 0 -3π/4 -0.707 -π/2 0 -π/4 0.707-0.707 010 π/4 0.707 π/2 01 3π/4 -0.7070.707 π 0 5π/4 -0.707 3π/2 0 7π/4 0.707-0.707 2π2π 10

13. The Cotangent Function Graph Find the values of cosine divided by sine. COT(X) 1/0 = DNE.707/.707 = 1 0/1 = 0 -.707/.707 = -1 -1/0 = DNE -.707/-.707 = 1 0/-1 = 0.707/-.707 = -1 1/0 = DNE.707/.707 = 1 0/1 = 0 -.707/.707 = -1 -1/0 = DNE -.707/-.707 = 1 0/-1 = 0.707/-.707 = -1 1/0 = DNE Plot the points.The errors are asymptotes. XCOS(X)SIN(X) -2π 10 -7π/4 0.707 -3π/2 01 -5π/4 -0.7070.707 -π-π 0 -3π/4 -0.707 -π/2 0 -π/4 0.707-0.707 010 π/4 0.707 π/2 01 3π/4 -0.7070.707 π 0 5π/4 -0.707 3π/2 0 7π/4 0.707-0.707 2π2π 10

14. The Cotangent Function Graph Domain: Range: Asymptotes All Reals except multiples of Pi. X = all multiples of Pi. All Reals.