Lesson 3: Right Triangle Definitions of Trig Functions
Estimated time: 25-30 minutes
Learning Objectives
By the end of this lesson, you will be able to:
- Define all six trig functions using opposite, adjacent, and hypotenuse
- Apply SOH-CAH-TOA to find trig values from a right triangle
- Find missing sides using the Pythagorean theorem and then evaluate trig functions
- Understand the relationship between unit circle and right triangle definitions
SOH-CAH-TOA
For an acute angle θ in a right triangle, the six trig functions are ratios of the sides relative to that angle.
SOH: sin θ = Opposite / Hypotenuse
CAH: cos θ = Adjacent / Hypotenuse
TOA: tan θ = Opposite / Adjacent
Reciprocal Functions:
csc θ = Hypotenuse / Opposite = 1/sin θ
sec θ = Hypotenuse / Adjacent = 1/cos θ
cot θ = Adjacent / Opposite = 1/tan θ
Labeling the Triangle
In a right triangle with acute angle θ:
- The hypotenuse is always the longest side, opposite the right angle.
- The opposite side is across from angle θ.
- The adjacent side is next to angle θ (but not the hypotenuse).
Important: the labels "opposite" and "adjacent" change depending on which acute angle you are considering.
Worked Examples
Example 1: Finding All Six Trig Values
A right triangle has legs 3 and 4 and hypotenuse 5. Find all six trig functions of the angle θ opposite the side of length 3.
Solution: Opposite = 3, Adjacent = 4, Hypotenuse = 5
sin θ = 3/5 cos θ = 4/5 tan θ = 3/4
csc θ = 5/3 sec θ = 5/4 cot θ = 4/3
Example 2: Using the Pythagorean Theorem First
In a right triangle, the side opposite θ is 7 and the hypotenuse is 25. Find cos θ.
Solution:
Step 1: Find adjacent: a² + 7² = 25², a² = 625 − 49 = 576, a = 24
Step 2: cos θ = adjacent/hypotenuse = 24/25
Example 3: Given One Trig Value, Find Another
If tan θ = 5/12 and θ is acute, find sin θ and cos θ.
Solution:
Step 1: Set up: Opposite = 5, Adjacent = 12. Find hypotenuse: h = √(25 + 144) = √169 = 13
Step 2: sin θ = 5/13, cos θ = 12/13
Connecting to the Unit Circle
The right triangle and unit circle definitions are consistent. When we place an acute angle in standard position and drop a perpendicular from the unit circle to the x-axis, we get a right triangle with hypotenuse 1. The opposite side equals sin θ and the adjacent side equals cos θ.
Cofunctions
In a right triangle, the two acute angles are complementary (sum to 90°). This creates the cofunction relationships:
sin θ = cos(90° − θ) and cos θ = sin(90° − θ)
tan θ = cot(90° − θ) and sec θ = csc(90° − θ)
Example 4: Cofunction
sin 40° = cos 50° (since 40 + 50 = 90).
tan 25° = cot 65° (since 25 + 65 = 90).
Check Your Understanding
1. In a right triangle with legs 8 and 15, find sin θ where θ is opposite the side of length 8.
2. If cos θ = 3/7, find sin θ (assume θ acute).
3. Express cos 72° as a function of a complementary angle.
Key Takeaways
- SOH-CAH-TOA defines sin, cos, tan using opposite, adjacent, hypotenuse.
- The reciprocal functions (csc, sec, cot) are the flipped ratios.
- Use the Pythagorean theorem to find missing sides before computing trig ratios.
- Cofunction identities: sin θ = cos(90° − θ) and similar pairs.