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Digital SAT · Right-Triangle Trigonometry

How to Solve Cofunctions and Complementary Angles on the Digital SAT

In a right triangle the two acute angles add to 90 degrees, and that makes sine and cosine trade places: the sine of an angle equals the cosine of its complement, and the other way around. So if the sine of A equals the cosine of B, then A and B are complementary and add to 90. Use it to find an unknown angle, or to read a sine value straight off a cosine, with no triangle drawn.

Written from Perfect1600’s analysis of every cofunctions and complementary angles question in our bank·Method checked against the current Bluebook test
per test
less than 1 per test
per test
typical difficulty
Medium
typical difficulty
practice questions
32
practice questions
How we counted

Frequency reflects how often this question type appears on a full-length Digital SAT. The difficulty mix reflects every question of this type across our bank.

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What the question bank shows

A + B = 90°
Sine of A = cosine of B

Sine and cosine swap for complementary angles, which add to 90.

one equation
No calculator

It reduces to complementary angles, not a computation.

the hurdle
Spot the shortcut

Seeing the cofunction link, rather than building a triangle, is the skill.

How to recognize cofunctions and complementary angles questions

  • A sine is set equal to a cosine, like the sine of A equals the cosine of B.
  • The question asks for an angle, or notes the angles are in a right triangle.
  • Complementary angles, adding to 90 degrees, are involved.
  • Choices are angle measures or trig values.

Why students miss these

Once you see the relationship it is short, but it is easy to miss. People try to build a triangle or compute values when the shortcut is that the two angles are complementary. Others set the angles equal instead of summing to 90. Spotting that sine and cosine swap for complementary angles turns the whole problem into one equation.

The step-by-step method

  1. 1

    Spot the sine-cosine pairing

    A sine set equal to a cosine signals complementary angles.

  2. 2

    Set the angles complementary

    If the sine of A equals the cosine of B, then A plus B equals 90 degrees.

  3. 3

    Solve for the unknown angle

    Use the sum of 90 to find the missing angle or its value.

  4. 4

    Read a value across functions

    The sine of an angle equals the cosine of its complement, so read one from the other.

Worked examples

Easy example
If cos⁡20∘=sin⁡θ\cos 20^\circ = \sin\theta for an acute angle θ\theta, what is the value of θ\theta, in degrees?
A
60
B
65
C
68
70

A: Incorrect. This does not satisfy 20+θ=9020 + \theta = 90.

B: Incorrect. This does not make the angles complementary.

C: Incorrect. This is close but does not equal the complement of 20∘20^\circ.

D: Correct. Since cos⁡20∘=sin⁡70∘\cos 20^\circ = \sin 70^\circ, θ=70\theta = 70.

Explanation

cos⁡20∘=sin⁡(90∘−20∘)=sin⁡70∘\cos 20^\circ = \sin(90^\circ - 20^\circ) = \sin 70^\circ, so θ=70\theta = 70.

Medium example
In a right triangle, acute angle AA satisfies sin⁡A=cos⁡((3A−10)∘)\sin A = \cos((3A - 10)^\circ), where AA is measured in degrees. What is the value of AA?
A
10
B
18
C
20
25

A: Incorrect. This does not satisfy A+(3A−10)=90A + (3A - 10) = 90.

B: Incorrect. This solves the equation incorrectly.

C: Incorrect. This sets 4A=804A = 80 rather than 100100.

D: Correct. Cofunctions give A+(3A−10)=90A + (3A - 10) = 90, so 4A=1004A = 100 and A=25A = 25.

Explanation

A+(3A−10)=90A + (3A - 10) = 90 gives 4A=1004A = 100, so A=25A = 25.

Hard example
In right triangle ABCABC, the right angle is at CC. If sin⁡A=817\sin A = \frac{8}{17}, what is the value of cos⁡B\cos B?
817\frac{8}{17}
B
815\frac{8}{15}
C
1517\frac{15}{17}
D
178\frac{17}{8}

A: Correct. Angles AA and BB are complementary, so cos⁡B=sin⁡A=817\cos B = \sin A = \frac{8}{17}.

B: Incorrect. This is tan⁡A\tan A, not cos⁡B\cos B.

C: Incorrect. This is cos⁡A\cos A (equivalently sin⁡B\sin B), not cos⁡B\cos B.

D: Incorrect. This inverts the sine ratio.

Explanation

Complementary angles give cos⁡B=sin⁡A=817\cos B = \sin A = \frac{8}{17}.

The common traps

PatternWhat it doesThe tell
Set angles equalMade the two angles equal instead of summing to 90.Sine equals cosine means complementary, not equal.
Built a triangleComputed sides when the cofunction shortcut applies.A sine equal to a cosine points straight to complementary angles.
Wrong complementSubtracted from 180 instead of 90.Complementary angles add to 90 degrees, not 180.

Try it: two real questions

Question 1easy

In a right triangle, AA and BB are the two acute angles. If sin⁡A=0.4\sin A = 0.4, what is the value of cos⁡B\cos B?

Question 2easy

If sin⁡35∘=cos⁡θ\sin 35^\circ = \cos\theta for an acute angle θ\theta, what is the value of θ\theta, in degrees?

Question 3 is ready when you are

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Often confused with

Right-Triangle Trigonometry
Trigonometric Ratios (SOH-CAH-TOA)
Circles
Degrees and Radians
Lines, Angles & Triangles
Angle Relationships

Related reading

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Common questions

What is the cofunction relationship on the SAT?

The sine of an angle equals the cosine of its complement, and the cosine of an angle equals the sine of its complement. It comes from the two acute angles of a right triangle adding to 90 degrees.

If sin A equals cos B, what do I know?

That A and B are complementary, so A plus B equals 90 degrees. You can solve for either angle from that one equation, no triangle needed.

Why do sine and cosine swap for complementary angles?

In a right triangle, the side opposite one acute angle is adjacent to the other, so the sine ratio for one angle is the cosine ratio for its complement.

Do I need a calculator for cofunction questions?

No. The relationship turns the problem into complementary angles adding to 90 degrees, solved with a short equation, not a computation.

Practice cofunctions and complementary angles the way it is tested

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