# Complementary

Complementary angles are two angles whose measures have a sum of 90°. Each angle is the complement of the other.

For a right triangle, the two non-right or oblique angles must be complementary.

In right triangle ABC above, ∠B = 90° and ∠A + ∠C = 90° so, the nonadjacent angles A and C are complements of each other.

You can determine the complement of a given angle by subtracting it from 90°. For example, the complement of 28° is 62° since 90° - 28° = 62°.

Example:

Look at the diagrams below and see if you can identify the complementary angles.

In right triangle ABC above, ∠C = 90° so angles A and B are complementary and, ∠A + 50° = 90°, then ∠A = 40°.

Since two angles do not need to be adjacent to be complementary, given enough information, we do not even need to have a diagram of complementary angles to figure them out.

Example:

Let ∠α and ∠θ be 2 angles that have the variable x in common. If ∠α and ∠θ are complementary where ∠α = (2x - 8)° and ∠θ = (x + 14)°, then

 (2x - 8)° + (x + 14)° = 90° 3x + 6° = 90° x = 28°

So, ∠α = (2×28 - 8)° = 48° and ∠θ = (28 + 14)° = 42°. Note that 48° + 42° = 90° verifies that ∠α and ∠θ are complementary.

In the study of Trigonometry, the sine value of an angle is equal to the cosine value of its complement. Also, the tangent value of an angle is equal to the cotangent value of its complement. So,

 sin(θ) = cos(90°-θ) and sin(90°-θ) = cos(θ) tan(θ) = cot(90°-θ) and tan(90°-θ) = cot(θ) where the complement for θ is 90°-θ

### Did you know?

Trigonometry is a branch of mathematics that studies the relationships between the side lengths and the angles of triangles. From the Greeks, trigonon means triangle, and metron means to measure.
The passage provides a detailed explanation of complementary angles, including examples and calculations, and introduces trigonometric relationships between complementary angles. It includes multiple examples that involve solving for angles, but the solutions are provided within the text.
multiple_questions_with_answers
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# Complementary

Complementary angles are two angles whose measures have a sum of 90°. Each angle is the complement of the other.

For a right triangle, the two non-right or oblique angles must be complementary.

In right triangle ABC above, ∠B = 90° and ∠A + ∠C = 90° so, the nonadjacent angles A and C are complements of each other.

You can determine the complement of a given angle by subtracting it from 90°. For example, the complement of 28° is 62° since 90° - 28° = 62°.

Example:

Look at the diagrams below and see if you can identify the complementary angles.

In right triangle ABC above, ∠C = 90° so angles A and B are complementary and, ∠A + 50° = 90°, then ∠A = 40°.

Since two angles do not need to be adjacent to be complementary, given enough information, we do not even need to have a diagram of complementary angles to figure them out.

Example:

Let ∠α and ∠θ be 2 angles that have the variable x in common. If ∠α and ∠θ are complementary where ∠α = (2x - 8)° and ∠θ = (x + 14)°, then

 (2x - 8)° + (x + 14)° = 90° 3x + 6° = 90° x = 28°

So, ∠α = (2×28 - 8)° = 48° and ∠θ = (28 + 14)° = 42°. Note that 48° + 42° = 90° verifies that ∠α and ∠θ are complementary.

In the study of Trigonometry, the sine value of an angle is equal to the cosine value of its complement. Also, the tangent value of an angle is equal to the cotangent value of its complement. So,

 sin(θ) = cos(90°-θ) and sin(90°-θ) = cos(θ) tan(θ) = cot(90°-θ) and tan(90°-θ) = cot(θ) where the complement for θ is 90°-θ

### Did you know?

Trigonometry is a branch of mathematics that studies the relationships between the side lengths and the angles of triangles. From the Greeks, trigonon means triangle, and metron means to measure.
user
The passage provides a detailed explanation of complementary angles, including examples and calculations, and introduces trigonometric relationships between complementary angles. It includes multiple examples that involve solving for angles, but the solutions are provided within the text.

TYPE: MULTIPLE_QUESTIONS_WITH_ANSWERS