TION 5
lify the following expressions as far as possible, without the use of a calculator:

Ask by Harper Marsh. in South Africa

Mar 30,2025

Question

TION 5 lify 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
$$
\sin \left(\beta+60^\circ\right)+\cos \left(\beta+150^\circ\right).
$$

**Step 1.** Use the identity $$\cos x = \sin\left(90^\circ - x\right)$$ to rewrite the cosine term:
$$
\cos \left(\beta+150^\circ\right) = \sin\left(90^\circ - (\beta+150^\circ)\right) = \sin\left(-\beta-60^\circ\right).
$$

**Step 2.** Use the odd property of the sine function, $$\sin(-\theta) = -\sin(\theta)$$:
$$
\sin\left(-\beta-60^\circ\right) = -\sin\left(\beta+60^\circ\right).
$$

**Step 3.** Substitute back into the original expression:
$$
\sin\left(\beta+60^\circ\right) + \cos\left(\beta+150^\circ\right) = \sin\left(\beta+60^\circ\right) - \sin\left(\beta+60^\circ\right) = 0.
$$

Thus, the simplified expression is 
$$
0.
$$
The expression simplifies to 0.

TION 5
lify the following expressions as far as possible, without the use of a calculator:

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