Right Triangle Trigonometry
📌 Introduction
Right triangle trigonometry is the study of relationships between the angles and sides of right triangles. These relationships form the foundation of trigonometry and have countless applications in mathematics, science, and engineering.
Pythagorean Theorem
The Formula:
\( a^2 + b^2 = c^2 \)
Where:
- \( a \) and \( b \) = legs of the right triangle
- \( c \) = hypotenuse (longest side, opposite the right angle)
Converse of Pythagorean Theorem:
If \( a^2 + b^2 = c^2 \) for sides of a triangle, then the triangle is a RIGHT triangle.
Testing triangle types:
- If \( a^2 + b^2 = c^2 \) → Right triangle
- If \( a^2 + b^2 > c^2 \) → Acute triangle
- If \( a^2 + b^2 < c^2 \) → Obtuse triangle
Special Right Triangles
45°-45°-90° Triangle:
An isosceles right triangle with two 45° angles
Side Ratios:
\( x : x : x\sqrt{2} \)
- Legs = \( x \)
- Hypotenuse = \( x\sqrt{2} \)
- If leg = 1, then hypotenuse = \( \sqrt{2} \)
30°-60°-90° Triangle:
A right triangle with angles of 30°, 60°, and 90°
Side Ratios:
\( x : x\sqrt{3} : 2x \)
- Short leg (opposite 30°) = \( x \)
- Long leg (opposite 60°) = \( x\sqrt{3} \)
- Hypotenuse (opposite 90°) = \( 2x \)
- If short leg = 1, sides are \( 1, \sqrt{3}, 2 \)
📝 Examples - Special Triangles:
Example 1: In a 45-45-90 triangle, if one leg = 5, find the hypotenuse
Hypotenuse = \( 5\sqrt{2} \)
Example 2: In a 30-60-90 triangle, if the short leg = 4, find the other sides
Long leg = \( 4\sqrt{3} \)
Hypotenuse = \( 2(4) = 8 \)
Trigonometric Ratios: Sin, Cos, and Tan
SOHCAHTOA - The Three Basic Ratios:
For angle θ in a right triangle:
\( \sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}} \)
\( \cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}} \)
\( \tan \theta = \frac{\text{Opposite}}{\text{Adjacent}} \)
Reciprocal Ratios: Csc, Sec, and Cot
The Three Reciprocal Ratios:
\( \csc \theta = \frac{1}{\sin \theta} = \frac{\text{Hypotenuse}}{\text{Opposite}} \)
\( \sec \theta = \frac{1}{\cos \theta} = \frac{\text{Hypotenuse}}{\text{Adjacent}} \)
\( \cot \theta = \frac{1}{\tan \theta} = \frac{\text{Adjacent}}{\text{Opposite}} \)
Alternative Expressions:
- \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)
- \( \cot \theta = \frac{\cos \theta}{\sin \theta} \)
- \( \sin^2 \theta + \cos^2 \theta = 1 \) (Pythagorean Identity)
The Unit Circle
Definition:
The unit circle is a circle with radius 1 centered at the origin of the coordinate plane.
Key Properties:
- Equation: \( x^2 + y^2 = 1 \)
- For any point \( (x, y) \) on the unit circle at angle θ:
- \( \cos \theta = x \) (x-coordinate)
- \( \sin \theta = y \) (y-coordinate)
- \( \tan \theta = \frac{y}{x} \)
Trig Values of Special Angles
Special Angles Table:
| Angle | sin | cos | tan |
|---|---|---|---|
| 0° or 0 | 0 | 1 | 0 |
| 30° or \( \frac{\pi}{6} \) | \( \frac{1}{2} \) | \( \frac{\sqrt{3}}{2} \) | \( \frac{\sqrt{3}}{3} \) |
| 45° or \( \frac{\pi}{4} \) | \( \frac{\sqrt{2}}{2} \) | \( \frac{\sqrt{2}}{2} \) | 1 |
| 60° or \( \frac{\pi}{3} \) | \( \frac{\sqrt{3}}{2} \) | \( \frac{1}{2} \) | \( \sqrt{3} \) |
| 90° or \( \frac{\pi}{2} \) | 1 | 0 | Undefined |
Reciprocal Function Values:
| Angle | csc | sec | cot |
|---|---|---|---|
| 30° | 2 | \( \frac{2\sqrt{3}}{3} \) | \( \sqrt{3} \) |
| 45° | \( \sqrt{2} \) | \( \sqrt{2} \) | 1 |
| 60° | \( \frac{2\sqrt{3}}{3} \) | 2 | \( \frac{\sqrt{3}}{3} \) |
Using Reference Angles
Finding Trig Values Using Reference Angles:
- Find the reference angle (acute angle to x-axis)
- Determine the quadrant of the original angle
- Find trig value for the reference angle
- Apply appropriate sign based on quadrant
Sign Rules by Quadrant:
- Quadrant I: All positive
- Quadrant II: Only sin positive
- Quadrant III: Only tan positive
- Quadrant IV: Only cos positive
Memory aid: "All Students Take Calculus"
Inverse Trigonometric Functions
Notation and Meaning:
Inverse functions find the angle when given the ratio:
- \( \sin^{-1} x \) or \( \arcsin x \) — finds angle whose sine is \( x \)
- \( \cos^{-1} x \) or \( \arccos x \) — finds angle whose cosine is \( x \)
- \( \tan^{-1} x \) or \( \arctan x \) — finds angle whose tangent is \( x \)
- \( \csc^{-1} x, \sec^{-1} x, \cot^{-1} x \) — reciprocal inverses
Range Restrictions:
- \( \sin^{-1} x \): Range is \( [-90°, 90°] \) or \( [-\frac{\pi}{2}, \frac{\pi}{2}] \)
- \( \cos^{-1} x \): Range is \( [0°, 180°] \) or \( [0, \pi] \)
- \( \tan^{-1} x \): Range is \( (-90°, 90°) \) or \( (-\frac{\pi}{2}, \frac{\pi}{2}) \)
Solving Trigonometric Equations
General Steps:
- Isolate the trigonometric function
- Use inverse function to find reference angle
- Find all solutions in the given interval
- Check your answers
📝 Examples - Solving Equations:
Example 1: Solve \( \sin \theta = 0.5 \) for \( 0° \leq \theta \leq 360° \)
\( \theta = \sin^{-1}(0.5) = 30° \) (reference angle)
Sin is positive in Quadrants I and II
Solutions: \( \theta = 30°, 150° \)
Example 2: Solve \( 2\cos x - 1 = 0 \)
\( 2\cos x = 1 \)
\( \cos x = \frac{1}{2} \)
\( x = 60° \) or \( \frac{\pi}{3} \)
Finding Side Lengths
Steps to Find a Side:
- Identify the known angle and known side
- Choose the appropriate trig ratio
- Set up the equation
- Solve for the unknown side
📝 Example - Finding Side Length:
In a right triangle, if the angle is 35° and the adjacent side is 10, find the opposite side.
Use: \( \tan 35° = \frac{\text{opposite}}{10} \)
\( \text{opposite} = 10 \times \tan 35° \)
\( \text{opposite} \approx 10 \times 0.700 = 7.0 \)
Finding Angle Measures
Steps to Find an Angle:
- Identify two known sides
- Choose the trig ratio involving those sides
- Use inverse trig function to find the angle
📝 Example - Finding Angle:
In a right triangle, the opposite side is 8 and the hypotenuse is 12. Find the angle.
Use: \( \sin \theta = \frac{8}{12} = \frac{2}{3} \)
\( \theta = \sin^{-1}\left(\frac{2}{3}\right) \)
\( \theta \approx 41.8° \)
Solving a Right Triangle
What Does "Solve" Mean?
To solve a right triangle means to find ALL six parts: 3 sides and 3 angles.
General Approach:
- You need at least one side and one angle (besides the 90°)
- OR two sides
- Use Pythagorean theorem for sides
- Use trig ratios for sides and angles
- Remember: angles in a triangle sum to 180°
📝 Complete Example - Solving a Triangle:
Given: angle A = 40°, hypotenuse = 15. Find all other parts.
Step 1: Find angle B
\( B = 90° - 40° = 50° \)
Step 2: Find side opposite to A
\( \sin 40° = \frac{a}{15} \)
\( a = 15 \sin 40° \approx 15(0.643) = 9.6 \)
Step 3: Find side adjacent to A
\( \cos 40° = \frac{b}{15} \)
\( b = 15 \cos 40° \approx 15(0.766) = 11.5 \)
Solution: A = 40°, B = 50°, C = 90°, a ≈ 9.6, b ≈ 11.5, c = 15
⚡ Quick Summary
| Concept | Formula/Key Idea |
|---|---|
| Pythagorean Theorem | \( a^2 + b^2 = c^2 \) |
| 45-45-90 Triangle | \( x : x : x\sqrt{2} \) |
| 30-60-90 Triangle | \( x : x\sqrt{3} : 2x \) |
| sin, cos, tan | SOH-CAH-TOA |
| Reciprocals | csc, sec, cot = 1/(sin, cos, tan) |
| Unit Circle | \( (\cos \theta, \sin \theta) \) |
- SOHCAHTOA: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent
- Special angles: 0°, 30°, 45°, 60°, 90° have exact trig values
- Use reference angles to find trig values in any quadrant
- Inverse trig functions find angles from ratios
- To solve a right triangle, find all 6 parts (3 sides, 3 angles)
📚 Pythagorean Identity & Related
Fundamental Identity:
\( \sin^2 \theta + \cos^2 \theta = 1 \)
Related Identities:
- \( 1 + \tan^2 \theta = \sec^2 \theta \)
- \( 1 + \cot^2 \theta = \csc^2 \theta \)
- \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)