Trigonometry

Trigonometry

Study Outline

Overview of Trigonometry

Explanation: Introduction to the fundamental concepts of trigonometry. What Will Be Taught: Understanding trigonometric ratios, the unit circle, and how to solve trigonometric equations. Why It’s Important: Trigonometry is essential for analyzing and solving problems involving angles, triangles, and periodic phenomena in various fields such as engineering, physics, and architecture.

Trigonometric Ratios

Explanation: Introduction to the basic trigonometric ratios. What Will Be Taught: Definitions of sine, cosine, and tangent, and their applications in right triangles. Why It’s Important: Trigonometric ratios are foundational tools for solving problems related to angles and distances in right-angled triangles.

The Unit Circle

Explanation: Understanding the unit circle and its significance in trigonometry. What Will Be Taught: The definition of the unit circle, how to determine the coordinates of points on the unit circle, and how to relate these coordinates to trigonometric functions. Why It’s Important: The unit circle is a powerful tool for understanding the behavior of trigonometric functions and for solving trigonometric equations.

Solving Trigonometric Equations

Explanation: Techniques for solving trigonometric equations. What Will Be Taught: Methods for solving basic and complex trigonometric equations, including the use of identities and inverse trigonometric functions. Why It’s Important: Solving trigonometric equations is essential for applications in various fields, including physics, engineering, and computer graphics.

Study Content

Overview of Trigonometry: Trigonometry is the branch of mathematics that studies the relationships between the sides and angles of triangles. It is widely used in various fields, including astronomy, engineering, and physics.

Trigonometric Ratios: The trigonometric ratios sine, cosine, and tangent are defined as ratios of the sides of a right-angled triangle.

  1. Sine (sin): The sine of an angle θ\thetaθ in a right-angled triangle is the ratio of the length of the opposite side to the length of the hypotenuse.

sin⁡(θ)=oppositehypotenuse\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}sin(θ)=hypotenuseopposite​

Example 1: In a right triangle with an angle θ=30∘\theta = 30^\circθ=30∘, and the hypotenuse is 10 units long, the opposite side is 5 units long:

sin⁡(30∘)=510=0.5\sin(30^\circ) = \frac{5}{10} = 0.5sin(30∘)=105​=0.5

  1. Cosine (cos): The cosine of an angle θ\thetaθ is the ratio of the length of the adjacent side to the length of the hypotenuse.

cos⁡(θ)=adjacenthypotenuse\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}cos(θ)=hypotenuseadjacent​

Example 2: In a right triangle with an angle θ=60∘\theta = 60^\circθ=60∘, and the hypotenuse is 10 units long, the adjacent side is 5 units long:

cos⁡(60∘)=510=0.5\cos(60^\circ) = \frac{5}{10} = 0.5cos(60∘)=105​=0.5

  1. Tangent (tan): The tangent of an angle θ\thetaθ is the ratio of the length of the opposite side to the length of the adjacent side.

tan⁡(θ)=oppositeadjacent\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}tan(θ)=adjacentopposite​

Example 3: In a right triangle with an angle θ=45∘\theta = 45^\circθ=45∘, where both the opposite and adjacent sides are equal in length (5 units):

tan⁡(45∘)=55=1\tan(45^\circ) = \frac{5}{5} = 1tan(45∘)=55​=1

The Unit Circle: The unit circle is a circle with a radius of 1 centered at the origin of a coordinate plane. It is used to define trigonometric functions for all angles.

  1. Coordinates on the Unit Circle: The coordinates of any point on the unit circle are given by (cos⁡(θ),sin⁡(θ))(\cos(\theta), \sin(\theta))(cos(θ),sin(θ)), where θ\thetaθ is the angle formed by the radius with the positive x-axis.

Example 4: The point corresponding to θ=90∘\theta = 90^\circθ=90∘ on the unit circle has coordinates (0,1)(0, 1)(0,1).

  1. Trigonometric Functions and the Unit Circle: Trigonometric functions can be defined for all angles using the unit circle, including those greater than 90∘90^\circ90∘ or less than 0∘0^\circ0∘.

Example 5: For θ=270∘\theta = 270^\circθ=270∘ on the unit circle, the coordinates are (0,−1)(0, -1)(0,−1), so:

sin⁡(270∘)=−1,cos⁡(270∘)=0\sin(270^\circ) = -1, \quad \cos(270^\circ) = 0sin(270∘)=−1,cos(270∘)=0

  1. Using the Unit Circle to Solve Problems: The unit circle allows us to extend the definitions of sine, cosine, and tangent to all angles, and to solve trigonometric equations.

Example 6: Solve sin⁡(θ)=22\sin(\theta) = \frac{\sqrt{2}}{2}sin(θ)=22​​:

    • From the unit circle, sin⁡(θ)=22\sin(\theta) = \frac{\sqrt{2}}{2}sin(θ)=22​​ at θ=45∘\theta = 45^\circθ=45∘ and θ=135∘\theta = 135^\circθ=135∘.

Solving Trigonometric Equations: Trigonometric equations can be solved using identities, the unit circle, and inverse trigonometric functions.

  1. Basic Trigonometric Equations: Equations involving trigonometric functions can often be solved by isolating the function and finding the angle that satisfies the equation.

Example 7: Solve cos⁡(θ)=0.5\cos(\theta) = 0.5cos(θ)=0.5:

    • From the unit circle, cos⁡(θ)=0.5\cos(\theta) = 0.5cos(θ)=0.5 at θ=60∘\theta = 60^\circθ=60∘ and θ=300∘\theta = 300^\circθ=300∘.
  2. Using Trigonometric Identities: Identities such as the Pythagorean identity sin⁡2(θ)+cos⁡2(θ)=1\sin^2(\theta) + \cos^2(\theta) = 1sin2(θ)+cos2(θ)=1 can be used to solve more complex equations.

Example 8: Solve sin⁡2(θ)+cos⁡2(θ)=1\sin^2(\theta) + \cos^2(\theta) = 1sin2(θ)+cos2(θ)=1 for θ\thetaθ:

    • Since the identity holds for all θ\thetaθ, any angle θ\thetaθ satisfies this equation.
  2. Inverse Trigonometric Functions: Inverse trigonometric functions can be used to find the angle corresponding to a given trigonometric value.

Example 9: Find θ\thetaθ such that tan⁡(θ)=1\tan(\theta) = 1tan(θ)=1:

    • θ=tan⁡−1(1)=45∘\theta = \tan^{-1}(1) = 45^\circθ=tan−1(1)=45∘.

Summary: This chapter covers the fundamental concepts of trigonometry, including trigonometric ratios, the unit circle, and methods for solving trigonometric equations. Understanding these concepts is essential for solving problems related to angles and triangles in various fields.