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