Trigonometry Basics: A Clear Guide

Lesson Overview

• What is Trigonometry?
• Trigonometric Functions
• Trigonometry Table
• Trigonometry Formulas
• Trigonometric Functions Graphs
• Trigonometric Equations and Identities 
• Trigonometric Functions Assessment

Trigonometry plays a key role, from calculating distances in navigation to designing ramps in architecture. Imagine you're building a ramp for a skateboard park. You can use trigonometry to determine the right angle for the ramp's slope based on the height and length.

What is Trigonometry?

Trigonometry focuses on the relationships between the angles and sides of triangles. The relationships between triangles' angles and sides are the main subject of trigonometry.

Trigonometric Functions

Trigonometric functions are mathematical functions that relate the angles of a right triangle to the lengths of   its sides.

Take this Quiz :

Trigonometry With Right Triangles Assessment Test

Trigonometry Table

In this, learn common angles and their trigonometry values. Let's get familiar with some common angles. 

• 0 degrees: This is a flat line.
• 30 degrees: Imagine slicing a pizza into 6 equal pieces. The angle at the tip of each slice is 30 degrees.
• 45 degrees: Think of the corner of a square. That's a 45-degree angle.
• 60 degrees: If you slice a pizza into 3 equal pieces, the angle at the tip of each slice is 60 degrees.
• 90 degrees: This is a right angle, like the corner of a book.

Take this Quiz :

Do you know your math? Trigonometry quiz

Trigonometry Formulas

Three main trigonometric functions: sine (sin), cosine (cos), and tangent (tan). Each one relates to a specific ratio in a right triangle.

Sine (sin): The sine of an angle is the ratio of the length of the opposite side (the side opposite to the angle) to the hypotenuse (the longest side of the triangle, opposite the right angle).

Example:

If the angle of elevation from the top of the building to the tree is 30∘ and the height of the building (opposite side) is 50 meters, one can calculate the distance along the ground (the base of the triangle) using the sine formula.

Cosine (cos): The cosine of an angle is the ratio of the length of the adjacent side (the side next to the angle) to the hypotenuse (the longest side of the triangle, opposite the right angle).

Example:

If the hypotenuse of a triangle is 100 meters and the angle of elevation from the ground to the top of a building is 45°, one can calculate the distance from the building (adjacent side) using the cosine formula.

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

Example: If the height of a building (opposite side) is 50 meters and the distance from the base of the building (adjacent side) is 100 meters, you can calculate the angle of elevation using the tangent formula.

SOH, CAH, TOH To Remember Trigonometry Formulas

Trigonometric Functions Graphs

The primary trigonometric functions we'll cover are sine (sin), cosine (cos), and tangent (tan).

 Sine Function (y = sin x)

• Shape: The sine graph is a smooth, continuous curve that oscillates between -1 and 1.
• Period: The period of the sine function is 2π, which means it repeats itself every 2π units along the x-axis.
• Symmetry: The sine function is an odd function, meaning it is symmetric about the origin.

Cosine Function (y = cos x)

• Shape: The cosine graph is also a smooth, continuous curve that oscillates between -1 and 1.
• Period: The period of the cosine function is 2π, similar to the sine function.
• Symmetry: The cosine function is an even function, meaning it is symmetric about the y-axis.

3. Tangent Function (y = tan x)

• Shape: The tangent graph consists of a series of vertical asymptotes and repeating curves.
• Period: The period of the tangent function is π.
• Asymptotes: The tangent function has vertical asymptotes at x = (π/2) + kπ, where k is an integer.

Take this Quiz:

Test yourself on various uses of Trigonometry quiz

Trigonometric Equations and Identities 

Trigonometric identities are equations that hold true for all values of the variables within their domain. They are essential for simplifying expressions and solving equations.

Fundamental Trigonometric Identities

Solved examples Of Trigonometric Functions 

Let's work through some examples. We'll use a right triangle with the following sides:

• Hypotenuse = 5
• Opposite side (to the angle we're interested in) = 3
• Adjacent side (to the angle we're interested in) = 4

Example 1: Finding the Sine of an Angle

Problem: Find sin(θ) where θ is the angle in the triangle.

Solution:

1. Identify the relevant sides: We need the opposite side (3) and the hypotenuse (5).
2. Apply the definition: sin(θ) = Opposite / Hypotenuse
3. Substitute the values: sin(θ) = 3 / 5
4. Simplify: sin(θ) = 0.6

Example 2: Finding the Cosine of an Angle

Problem: Find cos(θ) for the same angle.

Solution:

1. Identify the relevant sides: We need the adjacent side (4) and the hypotenuse (5).
2. Apply the definition: cos(θ) = Adjacent / Hypotenuse
3. Substitute the values: cos(θ) = 4 / 5
4. Simplify: cos(θ) = 0.8

Example 3: Finding the Tangent of an Angle

Problem: Find tan(θ).

Solution:

1. Identify the relevant sides: We need the opposite side (3) and the adjacent side (4).
2. Apply the definition: tan(θ) = Opposite / Adjacent
3. Substitute the values: tan(θ) = 3 / 4
4. Simplify: tan(θ) = 0.75

Example 4: Finding an Angle Using the Inverse Trigonometric Function

Problem: If sin(θ) = 0.5, find the angle θ.

Solution:

1. Use the inverse sine function (sin⁻¹): θ = sin⁻¹(0.5)
2. Use a calculator: θ = 30 degrees (or π/6 radians)

Trigonometric Functions Assessment

1. What is the sine of a 30-degree angle?
   
2. A right triangle has a hypotenuse of length 10 and an angle of 30 degrees. What is the length of the side opposite the 30-degree angle?
   
3. A Ferris wheel has a radius of 50 feet. The wheel makes one revolution every 30 seconds. Write a trigonometric function that models the height of a rider on the Ferris wheel as a function of time.