Exercises: Complementary angle relationship (sin x = cos(90-x)) - Digital SAT Math

A

Recall / Warm-Up

1.

What is $\\sin(30\degree)$ ?

A.

$\\dfrac{\\sqrt{3}}{2}$

B.

$\\dfrac{1}{2}$

C.

$\\dfrac{\\sqrt{2}}{2}$

D.

$1$

2.

Two acute angles in a right triangle are $A$ and $B$ . What must be true about $A + B$ ?

A.

$A + B = 180\degree$

B.

$A + B = 90\degree$

C.

$A + B = 45\degree$

D.

$A + B = 360\degree$

3.

In a right triangle, the right angle is at vertex $C$ . The two acute angles are $A$ and $B$ , where $B = 90\degree - A$ .
The side opposite $A$ has length 3 and the hypotenuse has length 5.

What is $\\cos(B)$ ?

A.

$\\dfrac{4}{5}$

B.

$\\dfrac{3}{5}$

C.

$\\dfrac{3}{4}$

D.

$\\dfrac{5}{3}$

B

Fluency Practice

1.

If $\\sin(x) = \\cos(35\degree)$ , find $x$ . Enter the value in degrees.

2.

If $\\cos(x) = \\sin(20\degree)$ , find $x$ . Enter the value in degrees.

3.

If $\\sin(x) = \\cos(x + 10\degree)$ , find $x$ . Enter the value in degrees.

4.

Which expression is equivalent to $\\sin(50\degree)$ ?

A.

$\\cos(50\degree)$

B.

$\\sin(40\degree)$

C.

$\\cos(40\degree)$

D.

$\\tan(50\degree)$

5.

If $\\sin(25\degree) = \\cos(y)$ , what is $y$ ?

A.

$155\degree$

B.

$65\degree$

C.

$25\degree$

D.

$115\degree$

C

Varied Practice

D

Word Problems / Application

1.

On a standardized test, a student encounters the following question: "For acute angle $\\theta$ , $\\sin(\\theta) = \\cos(28\degree)$ . What is the value of $\\theta$ ?"

What is the value of $\\theta$ in degrees?

2.

An architect designs a roof where a support beam makes angle $A$ with the horizontal. The complement of angle $A$ is $32\degree$ .

Which expression gives $\\sin(A)$ ?

A.

$\\sin(32\degree)$

B.

$\\cos(58\degree)$

C.

$\\cos(32\degree)$

D.

$\\tan(32\degree)$

3.

A calculator display shows that $\\sin(67\degree) \\approx 0.921$ .

1.

Using the complementary angle relationship, what is $\\cos(23\degree)$ ? Enter the decimal value.

2.

Without a calculator, determine which angle $x$ satisfies $\\sin(x) = \\cos(23\degree)$ .

A.

$23\degree$

B.

$67\degree$

C.

$157\degree$

D.

$113\degree$

E

Error Analysis

1.

Tyler solved the following problem:

"If $\\sin(x) = \\cos(35\degree)$ , find $x$ ."

Tyler's work: " $\\sin(x) = \\cos(35\degree)$ means $x = 35\degree$ because sine and cosine of the same angle are equal."

Tyler's answer: $x = 35\degree$

What mistake did Tyler make?

A.

Tyler should have used the tangent function instead.

B.

Tyler confused $\\sin(x) = \\cos(x)$ with the complementary identity — $\\sin(x) = \\cos(y)$ means $x + y = 90\degree$ , not $x = y$ .

C.

Tyler subtracted from 180° instead of 90°.

D.

Tyler's answer is actually correct.

2.

Sam wrote the following claim:

" $\\tan(x) = \\tan(90\degree - x)$ , so if $\\tan(x) = \\tan(y)$ , then $x + y = 90\degree$ ."

What is wrong with Sam's reasoning?

A.

Sam should have used sine, not tangent.

B.

Sam made an arithmetic error.

C.

The complementary angle relationship applies to sine and cosine only — $\\tan(x)$ does NOT equal $\\tan(90\degree - x)$ .

D.

Sam's equation is correctly set up.

F

Challenge / Extension

1.

If $\\sin(2x + 15\degree) = \\cos(3x - 10\degree)$ , find $x$ .

Verify that both resulting angles are between $0\degree$ and $90\degree$ before entering your answer.

2.

The word "cosine" literally means "complement's sine."

Explain in your own words, using a right triangle, why $\\sin(A) = \\cos(B)$ when $A$ and $B$ are the two acute angles of a right triangle.