SAT Practice Test - Math - Sine and Cosine Functions

SAT practice tests arranged by topic and difficulty level. In this section find tips and tactics for solving questions that focus on Sine and Cosine Functions.

_______________________


SAT Practice Test 2015.

In the xy-plane above, O is the center of the circle, and the measure of ∠AOB is $\pi/a$ radians. What is the value of a?

Answer:

Segment AO measures:

$(AO)^2=(\sqrt{3})^2+1^2$
$(AO)^2=3+1=4$
$AO=2$

Therefore sine of ∠AOB is 1/2. And the measure of ∠AOB is 30°, which is equal to $\pi/6$ radians.

Answer: a=6

_____________________________


SAT Practice Test. In a right triangle, one angle measures x°, where sin x°=4/5. What is cos(90°-x°)?

Answer:

In a right triangle, if one of the angles adjacent to the hypotenuse measures x, the other angle adjacent to the hypotenuse measures 90-x.
Sine of x can be calculated by the dividing the oposite side by the hypotenuse. It turns out that the oposite side to angle x is the adjacent side to angle 90-x. So
$sin(x)=cos(90-x)$.
$cos(90-x)=4/5$

_____________________________


SAT Practice Test. An architect drew the following sketch while designing a house roof. The dimensions shown are for the interior of the triangle.

The figure presents a triangle with a horizontal base. Labels are given to two sides and to two angles. The left side of the triangle is labeled 24 feet. The base of the triangle is labeled 32 feet. The two lower interior angles are labeled x degrees. A note under the figure says that the figure is not drawn to scale.

What is the value of  $cos(x)$?

Answer:

Because the triangle is isosceles, a perpendicular from the top vertex to the opposite side will bisect the base, creating two right triangles. The side of these triangles close to angle x will, therefore, measure 16ft. The hypotenuse of these triangles measure 24ft. Using these measures:

$cos(x)=16/24$
$cos(x)=2/3$

_____________________________


SAT Practice Test. It is given that $sin(x)=a$, where x is the radian measure of an angle and $\pi/2<x<\pi$.
If $sin(w)=-a$, which of the following could be the value of w?
A. $\pi-x$
B. $x-\pi$
C. $2\pi+x$
D. $x-2\pi$

Answer:

In the following figure, x is the angle $A\hat{O}X$, and w can be either $A\hat{O}W$ or $A\hat{O}W'$:

$A\hat{O}W=\pi+(\pi-x)=2\pi-x=-x$ (no match in the given answers).

$A\hat{O}W'=-(\pi-x)=x-\pi$

Answer: B

_____________________________


SAT Practice Test. Which of the following is equal to $sin(\frac{\pi}{5})$?

A.   $-cos(\frac{\pi}{5})$.

B.   $-sin(\frac{\pi}{5})$.

C.   $cos(\frac{3\pi}{10})$.

D.   $sin(\frac{7\pi}{10})$.

Answer:

Important relation between sine and cosine: $sin(x)=cos(\frac{\pi}{2}-x)$. Therefore:

$sin(\frac{\pi}{5})=cos(\frac{\pi}{2}-\frac{\pi}{5})$

$sin(\frac{\pi}{5})=cos(\frac{5\pi}{10}-\frac{2\pi}{10})$

$sin(\frac{\pi}{5})=cos(\frac{3\pi}{10})$

Answer: C

_____________________________