Problem 1 - Algebra 2 and Trigonometry

Question

In any right triangle, the acute angles are complementary. What is the relationship between the sine of the measure of an angle and the cosine of the measure of the complement of that angle? Justify your answer.

Step-by-Step Solution

Answer

The sine of an angle equals the cosine of its complement: \( \sin(\theta) = \cos(90^\circ - \theta) \).

Step 1: Understanding the Problem

We need to determine the relationship between the sine of an angle and the cosine of its complementary angle in a right triangle. Recall that complementary angles sum up to 90 degrees.

Step 2: Defining Complementary Angles

Let one of the acute angles in the triangle be \( \theta \), then the complement of this angle would be \( 90^\circ - \theta \).

Step 3: Using Sine and Cosine Definitions

The sine of an angle in a right triangle is defined as the ratio of the length of the opposite side to the hypotenuse. Similarly, the cosine of an angle is the ratio of the length of the adjacent side to the hypotenuse.

Step 4: Expressing in Terms of Sine and Cosine

For angle \( \theta \), sine is \( \sin\theta = \frac{\text{opposite}}{\text{hypotenuse}} \). For its complement \( 90^\circ - \theta \), cosine is \( \cos(90^\circ - \theta) = \frac{\text{adjacent to } \theta}{\text{hypotenuse}} \).

Step 5: Identifying Relationships

The side opposite \( \theta \) is adjacent to \( 90^\circ - \theta \), and both ratios involve the same hypotenuse. Thus, \( \sin(\theta) = \cos(90^\circ - \theta) \).

Step 6: Conclusion

The sine of an angle is equal to the cosine of its complementary angle in a right triangle, \( \sin(\theta) = \cos(90^\circ - \theta) \). This illustrates the co-function identity in trigonometry.

Key Concepts

Right Triangles

Right triangles are a fundamental concept in geometry. They have one angle that is exactly 90 degrees, known as the right angle. This type of triangle is unique because the two other angles are considered 'acute' angles, meaning they each measure less than 90 degrees. However, these two angles together always add up to 90 degrees, making them complementary angles.
In a right triangle:

• One angle measures exactly 90 degrees.
• The remaining two angles are complementary and sum up to 90 degrees.
• The longest side is called the hypotenuse, opposite the right angle.
• The other two sides, adjacent to the acute angles, are simply called the legs of the triangle.

Understanding right triangles is crucial because they form the basis of defining trigonometric functions like sine and cosine, which are used to describe relationships between the sides and angles.

Complementary Angles

Complementary angles are two angles whose measures add up to 90 degrees. In the context of right triangles, the two non-right angles are complementary because the sum of all angles in a triangle is always 180 degrees. Since one angle is already 90 degrees, the other two must be complementary. This is a key part of understanding trigonometric identities.
To visualize:

• If one angle is represented as \( \theta \), then its complement can be calculated as \( 90^\circ - \theta \).
• These relationships are essential in solving problems that involve sine and cosine functions.

Complementary angles enable the use of the sine and cosine relationship, illustrating how these functions transform based on angle complement values.

Sine and Cosine Relationship

The sine and cosine relationship is a central concept in trigonometry which involves defining the sine and cosine of angles, particularly in right triangles. These functions express how the lengths of sides relate to the angles in the triangle:

• For an angle \( \theta \), \( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \).
• For the complement \( 90^\circ - \theta \), the cosine is \( \cos(90^\circ - \theta) = \frac{\text{adjacent to } \theta}{\text{hypotenuse}} \).

What makes this relationship fascinating is the co-function identity: \( \sin(\theta) = \cos(90^\circ - \theta) \). This means the sine of any angle is equal to the cosine of its complement.
This identity helps in solving various trigonometric problems by allowing the conversion between sine and cosine functions, simplifying calculations especially when angles are complementary.