LESSON 12 SUM AND DIFFERENCE FORMULAS
The sum and difference formulas for the cosine function:
The sum and difference formulas for the sine function:
The sum and difference formulas for the tangent function:
Since , then we can find the exact value of the cosine, sine, and tangent of using the respective sum formula with and . Since and , then . Of course, we could have obtained this by converting to units of radians using the conversion factor . That is = = = = = .
The cosine of or :
=
= = …… (a)
=
= = …… (b)
The sine of or :
=
= = …… (c)
=
= = …… (d)
The tangent of or :
= =
= = = =
= = = =
= …… (e)
= =
= = …… (f)
Since is the reference angle for , , , and , then we will be able to find the exact value of the six trigonometric functions for these angles.
Since is the reference angle for , , , and , then we will be able to find the exact value of the six trigonometric functions for these angles.
Examples Use a reference angle to find the exact value of the six trigonometric functions of the following angles.
1.(This is the angle in units of degrees.)
The angle is in the II quadrant. The reference angle of the angle is the angle .
Since cosine is negative in the II quadrant and = by (b) above, then
= = =
= = =
= =
Since sine is positive in the II quadrant and = by (d) above, then
= =
= = =
= =
Since tangent is negative in the II quadrant and = by (f) above, then
= =
= = = =
= =
2.(This is the angle in units of radians.)
The angle is in the IV quadrant. The reference angle of the angle is the angle .
Since cosine is positive in the IV quadrant and = by (a) above, then
= =
= = =
= =
Since sine is negative in the IV quadrant and = by (c) above, then
= =
= = =
= = =
Since tangent is negative in the IV quadrant and = by (e) above, then
= =
= = = =
= =
3.(This is the angle in units of radians.)
The angle is in the III quadrant. The reference angle of the angle is the angle .
Since cosine is negative in the III quadrant and = by (a) above, then
= = =
= = =
= =
Since sine is negative in the III quadrant and = by (c) above, then
= =
= = =
= = =
Since tangent is positive in the III quadrant and = by (e) above, then
= =
= = = =
=
Since , then we can find the cosine, sine, and tangent of using the respective difference formula with and . Since and , then . Of course, we could have obtained this by converting to units of radians using the conversion factor . That is = = = = = .
The cosine of or :
=
= = …… (g)
=
= = …… (h)
The sine of or :
=
= = …… (i)
=
= = …… (j)
The tangent of or :
= =
= = = =
= = = …… (k)
= =
= = = …… (l)
NOTE: We also have that . So, you can find the exact value of the cosine, sine, and tangent of using the respective difference formula with and . Of course, you will obtain the same values that were obtained above.
Since is the reference angle for , , , and , then we will be able to find the exact value of the six trigonometric functions for these angles.
Since is the reference angle for , , , and , then we will be able to find the exact value of the six trigonometric functions for these angles.