Sum & Difference Identities Review

Front

What is the sum identity for cosine?

Back

The sum identity for cosine states that \( \cos(a + b) = \cos a \cos b - \sin a \sin b \).

Front

What is the difference identity for cosine?

Back

The difference identity for cosine states that \( \cos(a - b) = \cos a \cos b + \sin a \sin b \).

Front

What is the sum identity for sine?

Back

The sum identity for sine states that \( \sin(a + b) = \sin a \cos b + \cos a \sin b \).

Front

What is the difference identity for sine?

Back

The difference identity for sine states that \( \sin(a - b) = \sin a \cos b - \cos a \sin b \).

Front

How do you find the exact value of \( \cos(105^\circ) \) using sum or difference identities?

Back

You can express \( 105^\circ \) as \( 60^\circ + 45^\circ \) and use the cosine sum identity: \( \cos(105^\circ) = \cos(60^\circ)\cos(45^\circ) - \sin(60^\circ)\sin(45^\circ) \).

Front

What is the exact value of \( \sin(-15^\circ) \) using sum or difference identities?

Back

Using the identity, \( \sin(-15^\circ) = -\sin(15^\circ) \) and can be calculated using \( \sin(15^\circ) = \sin(45^\circ - 30^\circ) \).

Front

What is the value of \( \cos(20^\circ)\cos(40^\circ) - \sin(20^\circ)\sin(40^\circ) \)?

Back

Using the cosine difference identity, this expression simplifies to \( \cos(20^\circ + 40^\circ) = \cos(60^\circ) = \frac{1}{2} \).

Front

What is the exact value of \( \cos(105^\circ) \) using the cosine sum identity?

Back

Using the identity, \( \cos(105^\circ) = \cos(60^\circ + 45^\circ) = \cos(60^\circ)\cos(45^\circ) - \sin(60^\circ)\sin(45^\circ) = \frac{ \sqrt{2}-\sqrt{6}}{4} \).

Front

What is the relationship between sine and cosine for complementary angles?

Back

For complementary angles, \( \sin(90^\circ - x) = \cos(x) \) and \( \cos(90^\circ - x) = \sin(x) \).

Front

What is the value of \( \sin(30^\circ) \)?

Back

The value of \( \sin(30^\circ) \) is \( \frac{1}{2} \).