Question

verify the identity
$(5\sin\theta + 5\cos\theta)^2 = 25 + 25\sin2\theta$
begin by working with the left side. square $(5\sin\theta + 5\cos\theta)$.
(simplify your answer.)
apply a pythagorean identity and simplify the expression obtained in the previous step.
(simplify your answer.)
the expression from the previous step then simplifies to $25 + 25\sin2\theta$ using what?
a. half - angle formula
b. power - reducing formulas
c. sum and difference formula
d. double - angle formula

Explanation:

Step1: Square the binomial

$$\begin{align*} (5\sin\theta + 5\cos\theta)^2&=(5\sin\theta)^2+2\times5\sin\theta\times5\cos\theta+(5\cos\theta)^2\\ & = 25\sin^{2}\theta+50\sin\theta\cos\theta + 25\cos^{2}\theta \end{align*}$$

Step2: Apply Pythagorean identity \(\sin^{2}\theta+\cos^{2}\theta = 1\)

$$\begin{align*} 25\sin^{2}\theta+50\sin\theta\cos\theta + 25\cos^{2}\theta&=25(\sin^{2}\theta+\cos^{2}\theta)+50\sin\theta\cos\theta\\ &=25\times1 + 50\sin\theta\cos\theta\\ &=25+50\sin\theta\cos\theta \end{align*}$$

Step3: Apply double - angle formula \(\sin2\alpha=2\sin\alpha\cos\alpha\) (here \(\alpha = \theta\), so \(\sin2\theta=2\sin\theta\cos\theta\) or \(25\sin2\theta = 50\sin\theta\cos\theta\))

$$ 25+50\sin\theta\cos\theta=25 + 25\sin2\theta $$

Answer:

The first blank: \(25\sin^{2}\theta+50\sin\theta\cos\theta + 25\cos^{2}\theta\)

The second blank: \(25 + 50\sin\theta\cos\theta\)

For the multiple - choice question: D. Double - Angle Formula