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Chapter 6: Problem 1

Fill in the blank. Sine and _____ are cofunctions.

Short Answer

Expert verified
cosine

Step by step solution

01

- Identify Cofunctions

Cofunctions are pairs of trigonometric functions where the function of one angle equals the cofunction of the complementary angle.
02

- Identify Sine's Cofunction

The cofunction of sine is cosine. This is because \(\text{sin}(90^{\circ} - x) = \text{cos}(x)\) and vice versa.
03

- Fill in the Blank

Now that it's identified that sine and cosine are cofunctions, fill in the blank: Sine and cosine are cofunctions.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Sine
The sine function is one of the fundamental trigonometric functions. It relates a right-angled triangle's angle to the ratio of the length of the side opposite the angle to the hypotenuse.
To be clear, if we have an angle \(\theta\) in a right-angled triangle, the sine of this angle (written as \(\text{sin}(\theta)\)) is:

\[ \text{sin}(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \]

This function is very useful in various areas of mathematics and physics. Understanding sine can greatly help in solving problems involving waves, circles, and oscillations.
Next up, cosine!
Cosine and Its Relationship with Sine
Just like sine, the cosine function is a primary trigonometric function. It relates a right-angled triangle's angle to the ratio of the length of the adjacent side to the hypotenuse. For an angle \(\theta\), cosine (written as \(\text{cos}(\theta)\)) is defined as:

\[ \text{cos}(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \]

Cosine is deeply related to sine through what we call cofunctions. Cofunctions are pairs of trigonometric functions that complete each other. Specifically, \(\text{sin}(90^{\triangle} - x) = \text{cos}(x)\).
This makes sine and cosine a complementary pair, interesting and useful in various calculations.
Complementary Angles and Cofunctions
Complementary angles are two angles that add up to 90 degrees. When angles are complementary, the trigonometric function of one angle becomes the cofunction of the other. For example:
  • The sine of one angle equals the cosine of the complementary angle: \(\text{sin}(90^{\triangle} - x) = \text{cos}(x)\).
  • Similarly, the cosine of one angle equals the sine of the complementary angle: \(\text{cos}(90^{\triangle} - x) = \text{sin}(x)\).

This concept is critical in trigonometry and helps with solving various proof and calculation problems.
So remember, sine and cosine are cofunctions because they are linked through complementary angles, making them a powerful tool for your math toolkit!

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