If θ is an acute angle and sin2θ=a, then sinθ+cosθ equals

1976 AMC 12 Problem 17

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17.

\theta

is an acute angle and

\sin2\theta=a,

then

\sin\theta+\cos\theta

equals

\sqrt{a+1}

(\sqrt2-1)a+1

\sqrt{a+1}-\sqrt{a^2-a}

\sqrt{a+1}+\sqrt{a^2-a}

\sqrt{a+1}+a^2-a

Square the expression

\sin\theta+\cos\theta

Use

2\sin\theta\cos\theta=\sin2\theta

and the fact that the desired sum is positive

We have

\begin{aligned} (\sin\theta+\cos\theta)^2 &=\sin^2\theta+\cos^2\theta\\ &\quad+2\sin\theta\cos\theta\\ &=1+a. \end{aligned}

Both sine and cosine are positive for acute

\theta,

so the sum is

\sqrt{a+1}.

Therefore, the correct answer is A.