Comprehensive Trigonometry Notes: Sine, Cosine, and Tangent

Note: These notes cover the fundamental concepts of trigonometry, focusing on sine, cosine, and tangent functions. Various examples, solution methods, and a quiz are included to help reinforce your understanding.

1. Introduction to Trigonometry

Trigonometry is a branch of mathematics that studies the relationships between the sides and angles of triangles. The word "trigonometry" comes from the Greek words "trigonon" (triangle) and "metron" (measure).

1.1 The Right Triangle

In a right triangle, the three basic trigonometric functions are defined as ratios of the sides:

Sine (sin): The ratio of the opposite side to the hypotenuse
sin θ = opposite / hypotenuse

Cosine (cos): The ratio of the adjacent side to the hypotenuse
cos θ = adjacent / hypotenuse

Tangent (tan): The ratio of the opposite side to the adjacent side
tan θ = opposite / adjacent = sin θ / cos θ

1.2 Mnemonic: SOH-CAH-TOA

To remember the definitions of sine, cosine, and tangent:

SOH: Sine = Opposite / Hypotenuse
CAH: Cosine = Adjacent / Hypotenuse
TOA: Tangent = Opposite / Adjacent

1.3 The Unit Circle

The unit circle is a circle with a radius of 1 unit, centered at the origin of a coordinate plane. It provides another way to understand and define trigonometric functions.

For any angle θ on the unit circle:
sin θ = y-coordinate
cos θ = x-coordinate
tan θ = y-coordinate / x-coordinate = sin θ / cos θ

Move the slider to see how points on the unit circle correspond to sine and cosine values:

Angle: 0°

sin(θ) =

0.0000

cos(θ) =

1.0000

tan(θ) =

0.0000

2. Special Angles and Values

Certain angles have exact trigonometric values that are important to memorize:

Angle (degrees)	Angle (radians)	sin θ	cos θ	tan θ
0°	0	0	1	0
30°	π/6	1/2	√3/2	1/√3
45°	π/4	1/√2	1/√2	1
60°	π/3	√3/2	1/2	√3
90°	π/2	1	0	Undefined
180°	π	0	-1	0
270°	3π/2	-1	0	Undefined
360°	2π	0	1	0

2.1 Angle Measure: Degrees vs. Radians

There are two common ways to measure angles in trigonometry:

Degrees: A full circle is 360 degrees (360°)
Radians: A full circle is 2π radians (approximately 6.28 radians)

Conversion formulas:
Degrees to Radians: θ_rad = θ_deg × (π/180)
Radians to Degrees: θ_deg = θ_rad × (180/π)

3. Types of Trigonometric Problems and Solution Methods

3.1 Finding Missing Sides in Right Triangles

Problem: In a right triangle, one angle is 30° and the hypotenuse is 10 cm. Find the lengths of the other two sides.

Solution Method 1: Using SOH-CAH-TOA

We know one angle (30°) and the hypotenuse (10 cm)
For the opposite side: sin(30°) = opposite / hypotenuse
opposite = hypotenuse × sin(30°) = 10 × (1/2) = 5 cm
For the adjacent side: cos(30°) = adjacent / hypotenuse
adjacent = hypotenuse × cos(30°) = 10 × (√3/2) = 5√3 ≈ 8.66 cm

Solution Method 2: Using the Pythagorean Theorem

We already found one side to be 5 cm
Using the Pythagorean theorem: a² + b² = c²
5² + b² = 10²
25 + b² = 100
b² = 75
b = √75 = 5√3 ≈ 8.66 cm

Answer: The opposite side is 5 cm and the adjacent side is 5√3 ≈ 8.66 cm.

3.2 Finding Missing Angles in Right Triangles

Problem: In a right triangle, the opposite side is 8 units and the hypotenuse is 17 units. Find the measure of angle θ.

Solution Method 1: Using Inverse Sine

We know the opposite side (8) and the hypotenuse (17)
sin(θ) = opposite / hypotenuse = 8/17
θ = sin^-1(8/17) ≈ 28.07°

Solution Method 2: Using Inverse Cosine

First, we need to find the adjacent side using the Pythagorean theorem
adjacent² + 8² = 17²
adjacent² + 64 = 289
adjacent² = 225
adjacent = 15
Now we can use cos(θ) = adjacent / hypotenuse = 15/17
θ = cos^-1(15/17) ≈ 28.07°

Solution Method 3: Using Inverse Tangent

We found the adjacent side to be 15 units
tan(θ) = opposite / adjacent = 8/15
θ = tan^-1(8/15) ≈ 28.07°

Answer: The angle θ is approximately 28.07°.

3.3 Solving Real-World Applications

Problem: From a point 30 meters away from the base of a building, the angle of elevation to the top of the building is 40°. Find the height of the building.

Solution:

Draw a right triangle with the building as the opposite side, the ground distance as the adjacent side, and the line of sight as the hypotenuse
We know the adjacent side (30 m) and the angle (40°)
We need to find the opposite side (height)
tan(40°) = opposite / adjacent
tan(40°) = height / 30
height = 30 × tan(40°) = 30 × 0.8391 ≈ 25.17 meters

Answer: The height of the building is approximately 25.17 meters.

Problem: From a lighthouse 45 meters tall, the angle of depression to a ship is 25°. How far is the ship from the base of the lighthouse?

Solution:

The angle of depression is the angle below the horizontal, which forms the same angle with the vertical as the angle of elevation
We know the opposite side (45 m) and the angle (25°)
We need to find the adjacent side (distance)
tan(25°) = opposite / adjacent
tan(25°) = 45 / distance
distance = 45 / tan(25°) = 45 / 0.4663 ≈ 96.50 meters

Answer: The ship is approximately 96.50 meters from the base of the lighthouse.

3.4 Trigonometric Identities and Equations

Basic Trigonometric Identities:
sin²θ + cos²θ = 1
tan θ = sin θ / cos θ
cot θ = cos θ / sin θ = 1 / tan θ
sec θ = 1 / cos θ
csc θ = 1 / sin θ

Problem: Verify the identity: sin θ / (1 - cos θ) = (1 + cos θ) / sin θ

Solution Method 1: Working with the Left Side

Start with the left side: sin θ / (1 - cos θ)
Multiply numerator and denominator by (1 + cos θ):
[sin θ × (1 + cos θ)] / [(1 - cos θ) × (1 + cos θ)]
The denominator simplifies: (1 - cos²θ) = sin²θ (using the identity sin²θ + cos²θ = 1)
So we get: [sin θ × (1 + cos θ)] / sin²θ
Simplifying: (1 + cos θ) / sin θ, which is the right side

Solution Method 2: Working with Both Sides

Cross multiply to get: sin²θ = (1 - cos θ)(1 + cos θ)
Right side expands to: 1 - cos²θ
Using the identity sin²θ + cos²θ = 1, we get sin²θ = 1 - cos²θ
Thus, the identity is verified

Answer: The identity is verified.

Problem: Solve the equation: 2sin²θ - sin θ - 1 = 0 for 0° ≤ θ < 360°

Solution:

Let u = sin θ, so our equation becomes: 2u² - u - 1 = 0
Using the quadratic formula: u = [-(-1) ± √((-1)² - 4(2)(-1))] / (2(2))
u = (1 ± √(1 + 8)) / 4 = (1 ± 3) / 4
u = 1/4 or u = -1
So, sin θ = 1/4 or sin θ = -1
For sin θ = 1/4:
- θ = sin^-1(1/4) ≈ 14.5°
- θ = 180° - 14.5° = 165.5°
For sin θ = -1:
- θ = 270°

Answer: The solutions are θ ≈ 14.5°, 165.5°, and 270°.

3.5 Graphing Trigonometric Functions

Standard Forms:
y = A sin(Bx - C) + D
y = A cos(Bx - C) + D
y = A tan(Bx - C) + D
Where:
|A| = amplitude (for sine and cosine)
2π/|B| = period
C/B = phase shift
D = vertical shift

Problem: Graph y = 2sin(3x - π/2) + 1 and identify its key features.

Solution:

Identify the components:
- A = 2 (amplitude)
- B = 3 (affects period)
- C = π/2 (affects phase shift)
- D = 1 (vertical shift)
Calculate the period: 2π/|B| = 2π/3
Calculate the phase shift: C/B = (π/2)/3 = π/6
The function oscillates between -2+1 = -1 and 2+1 = 3

Key Features: