UESTION 5
implify the following expressions as far as possible, without the use of a calculator:

Ask by Young Li. in South Africa

Mar 30,2025

Question

UESTION 5 implify the following expressions as far as possible, without the use of a calculator: $$ \sin \left(\beta+60^{\circ}\right)+\cos \left(\beta+150^{\circ}\right) $$

UpStudy AI · Accepted Answer

We start with the expression
$$
\sin\left(\beta+60^\circ\right) + \cos\left(\beta+150^\circ\right).
$$

Notice that we can rewrite the second term by expressing the cosine function in terms of sine. Recall the identity:
$$
\cos(\theta + 90^\circ) = -\sin \theta.
$$
We write
$$
\beta+150^\circ = (\beta+60^\circ) + 90^\circ.
$$

Apply the identity:
$$
\cos\left((\beta+60^\circ)+90^\circ\right) = -\sin\left(\beta+60^\circ\right).
$$

Thus, the expression becomes:
$$
\sin\left(\beta+60^\circ\right) + \left[-\sin\left(\beta+60^\circ\right)\right].
$$

Simplify by combining like terms:
$$
\sin\left(\beta+60^\circ\right) - \sin\left(\beta+60^\circ\right) = 0.
$$

The simplified form is:
$$
0.
$$
The expression simplifies to 0.

UESTION 5
implify the following expressions as far as possible, without the use of a calculator:

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