Super-Hard SAT Math Question 85 | Free SAT Prep | Aniko

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Question 85·200 Super-Hard SAT Math Questions·Geometry and Trigonometry

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Which choice is the value of the expression below for all values of $x$ ?

\sin\big((20+x)^{\circ}\big)\cos\big((70-x)^{\circ}\big) +\sin\big((70-x)^{\circ}\big)\cos\big((20+x)^{\circ}\big) +\cos\big((20+x)^{\circ}\big)\cos\big((70-x)^{\circ}\big) -\sin\big((20+x)^{\circ}\big)\sin\big((70-x)^{\circ}\big)

When you see products of sines and cosines, try to group terms into angle-sum identities: $\sin A\cos B+\sin B\cos A=\sin(A+B)$ and $\cos A\cos B-\sin A\sin B=\cos(A+B)$ . Also check whether $A+B$ is a special angle (like $90^\circ$ ) so you can evaluate exactly.

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Check the sum of the angles

Set $A=(20+x)^\circ$ and $B=(70-x)^\circ$ . Compute $A+B$ .

Look for $\sin(A+B)$

Do any terms match $\sin A\cos B+\sin B\cos A$ ? If so, rewrite that part as $\sin(A+B)$ .

Look for $\cos(A+B)$

Do any terms match $\cos A\cos B-\sin A\sin B$ ? If so, rewrite that part as $\cos(A+B)$ .

Evaluate at $A+B=90^\circ$

After rewriting, you should have $\sin(A+B)+\cos(A+B)$ . Substitute $A+B=90^\circ$ and use the exact trig values for $90^\circ$ .

Desmos Guide

Set Desmos to degrees

In Desmos settings, set the angle unit to Degrees.

Define the expression

Enter

f(x)=sin(20+x)cos(70-x)+sin(70-x)cos(20+x)+cos(20+x)cos(70-x)-sin(20+x)sin(70-x)

(If needed, use degree symbols: sin((20+x)°) etc.).

Test several values of x

Make a table for $x$ (for example $0$ , $10$ , $25$ ) and observe that $f(x)$ stays constant. Match that constant to an answer choice (without relying on rounding).

Step-by-step Explanation

Relate the angles

Let $A=(20+x)^\circ$ and $B=(70-x)^\circ$ . Then

A+B=(20+x)^\circ+(70-x)^\circ=90^\circ.

Recognize a sine angle-sum pattern

The first two terms match the identity

\sin A\cos B+\sin B\cos A=\sin(A+B).

\sin(A)\cos(B)+\sin(B)\cos(A)=\sin(A+B).

Recognize a cosine angle-sum pattern

The last two terms match the identity

\cos A\cos B-\sin A\sin B=\cos(A+B).

\cos(A)\cos(B)-\sin(A)\sin(B)=\cos(A+B).

Evaluate using $A+B=90^\circ$

Putting the parts together,

\sin(A+B)+\cos(A+B)=\sin(90^\circ)+\cos(90^\circ)=1+0=1.

So the correct choice is $1$ .

Strategy

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Desmos Guide

Step-by-step Explanation

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Strategy

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Desmos Guide

Step-by-step Explanation