Right Triangle Trigonometry: Foundations

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Video introduction to right triangle trigonometry

Understanding Right Triangle Trigonometry

Trigonometry is the study of relationships between angles and sides in triangles. Right triangle trigonometry focuses specifically on right triangles (triangles with one 90° angle).

The Trigonometric Ratios

For a given angle in a right triangle, there are three main trigonometric ratios:

[Diagram of right triangle with sides labeled Opposite, Adjacent, and Hypotenuse relative to angle θ]

Ratio	Definition	Abbreviation
Sine	Opposite / Hypotenuse	sin θ = O/H
Cosine	Adjacent / Hypotenuse	cos θ = A/H
Tangent	Opposite / Adjacent	tan θ = O/A

Remember: SOH-CAH-TOA

Sine = Opposite / Hypotenuse
Cosine = Adjacent / Hypotenuse
Tangent = Opposite / Adjacent

Special Right Triangles

Two special right triangles have consistent ratios you should memorize:

45°-45°-90° Triangle

[Diagram of 45-45-90 triangle with legs = 1, hypotenuse = √2]

In a 45°-45°-90° triangle, the two legs are equal, and the hypotenuse is √2 times longer than each leg.

If leg = x, then hypotenuse = x√2

30°-60°-90° Triangle

[Diagram of 30-60-90 triangle with sides 1, √3, 2]

In a 30°-60°-90° triangle:

The side opposite 30° is the shortest side (x)
The side opposite 60° is x√3
The hypotenuse is 2x

Strategies for Solving Trigonometry Problems

Identify what you know: Which sides and angles are given?
Determine what you need to find: Is it a side length or angle measure?
Choose the appropriate ratio: Use SOH-CAH-TOA to select sine, cosine, or tangent.
Set up the equation: Write the trigonometric equation based on the ratio.
Solve for the unknown: Use algebra to solve for the missing value.
Check if special triangles apply: 45-45-90 or 30-60-90 triangles can simplify calculations.

Practice Question

[Diagram of right triangle with angle 35°, adjacent side = 8, opposite side = x]

In the right triangle above, the angle measures 35° and the side adjacent to this angle measures 8 units. What is the length of the side opposite the 35° angle, to the nearest tenth?

A) 4.6

B) 5.6

C) 6.8

D) 9.8

Explanation

We're given:

Angle θ = 35°
Adjacent side (A) = 8
Opposite side (O) = x (what we're solving for)

Since we have the adjacent side and want the opposite side, we'll use the tangent function:

tan θ = opposite / adjacent

tan 35° = x / 8

x = 8 × tan 35°

Using a calculator (make sure it's in degree mode):

tan 35° ≈ 0.7002

x ≈ 8 × 0.7002 ≈ 5.6016

Rounded to the nearest tenth: x ≈ 5.6

The correct answer is B) 5.6.

Common mistakes:

Using the wrong trigonometric function (sine or cosine instead of tangent)
Forgetting to put the calculator in degree mode
Rounding too early in the calculation

Additional Tips

Always check if your calculator is in degree mode when working with angles in degrees.
For angles not in special triangles, you'll need to use the trigonometric functions on your calculator.
The Pythagorean theorem (a² + b² = c²) often works together with trigonometry in right triangle problems.
When solving for an angle, use the inverse trigonometric functions (sin⁻¹, cos⁻¹, tan⁻¹).