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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