Question

For each triangle below, write an equation relating the reference angle with the two side lengths of the right triangle.

a) Provide equations for all triangles.
b) Provide equations for only the first triangle.
c) Provide equations for only the second triangle.
d) Provide equations for both triangles.

Answer 1

The student's question involves using trigonometric ratios—sine, cosine, and tangent—to write equations that relate the reference angle to the side lengths in right triangles and also leveraging the Pythagorean theorem to express the relationship between the sides of the triangle.

Step-by-step explanation:

The student is asking to relate the reference angle to the side lengths of right triangles using trigonometric ratios. The main trigonometric ratios include sine, cosine, and tangent which are functions of an angle and are fundamental in trigonometry. These ratios relate the lengths of the sides in a right triangle to its angles.

For a right triangle with a reference angle θ (theta), adjacent side x, opposite side y, and hypotenuse h, the equations are as follows:

• sine(θ) = y / h
• cosine(θ) = x / h
• tangent(θ) = y / x

Additionally, according to the Pythagorean theorem, the relationship between the lengths of the sides of a right triangle is:

a² + b² = c²

This can also be expressed as:

c = √(a² + b²)

Together, these form the fundamental equations for working with right triangles in trigonometry.