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Angle Sum and Difference Identities

Rule

Angle Sum and Difference Identities

To evaluate trigonometric functions of the sum of two angles, the following identities can be applied.

There are also similar identities for the difference of two angles.

These identities are useful to find the exact value of the sine, cosine, or tangent at a given angle. For example, the exact value of can be found by using the first formula. The first step is, in this case, to rewrite as Below, a proof of the first identity is shown. The rest of the identities can be proven using similar reasoning.

Proof

Proof

Let be a right triangle with hypotenuse and an acute angle with measure

Right triangle with hypotenuse length of 1 and acute angle x+y

By definition, the sine of an angle is the quotient between the length of the opposite side and the hypotenuse. Next, by drawing a ray, can be divided into two angles with measures and Let be a point on this ray such that and are right triangles.

Consider By calculating the sine and cosine of the legs of this triangle can be rewritten. Consider Knowing that the sine of can be used to write in terms of and Let be the point of intersection between and Notice that by the Vertical Angles Theorem.

By the Third Angle Theorem, it is known that Therefore,

Now, let be a point on such that This implies that In addition, is a right triangle and then can be calculated using

Finally, by the Segment Addition Postulate, is equal to the sum of and All these lengths have been rewritten in terms of the sine and cosine of and