Trigonometric Functions Study Guide

What are Trigonometric Functions?

Trigonometric functions are mathematical functions that relate the angles of a right-angled triangle to the lengths of its sides. The primary trigonometric functions are sine, cosine, and tangent. These functions are widely used in geometry, physics, engineering, and other fields.

Main Trigonometric Functions

The main trigonometric functions are:

• Sine (sinθ): The ratio of the length of the side opposite the given angle to the length of the hypotenuse in a right-angled triangle.
• Cosine (cosθ): The ratio of the length of the adjacent side to the length of the hypotenuse in a right-angled triangle.
• Tangent (tanθ): The ratio of the length of the side opposite the given angle to the length of the adjacent side in a right-angled triangle.

Trigonometric Identities

Trigonometric functions have several identities that are useful for simplifying expressions and solving equations. Some important identities include:

• Pythagorean Identity: sin^2θ + cos^2θ = 1
• Reciprocal Identities: cscθ = 1/sinθ, secθ = 1/cosθ, cotθ = 1/tanθ
• Quotient Identities: tanθ = sinθ/cosθ, cotθ = cosθ/sinθ
• Even-Odd Identities: sin(-θ) = -sinθ, cos(-θ) = cosθ, tan(-θ) = -tanθ

Graphs of Trigonometric Functions

The graphs of trigonometric functions exhibit periodic behavior. The sine and cosine functions, for example, have a period of 2π and oscillate between -1 and 1. The tangent function has vertical asymptotes at odd multiples of π/2.

Applications of Trigonometric Functions

Trigonometric functions are used in various real-world applications, including measuring heights and distances, analyzing periodic phenomena (such as sound waves and electrical signals), and designing structures and machinery.

Practice Problems

Test your understanding with the following practice problems:

1. Calculate the value of sin(60°).
2. Determine the values of cos(π/3) and tan(π/4).
3. Simplify the expression sin^2θ + cos^2θ.
4. Graph the function y = 2sin(x) in the interval [0, 2π].
5. Find the value of θ in the equation tanθ = 1.

Good luck with your trigonometric functions study!