5.4Right Triangle Trigonometry
Use SOH-CAH-TOA to define the six trig ratios in a right triangle. Solve for missing sides and angles, apply inverse trig functions, and solve angles of elevation and depression problems.
Right triangle trig is the bridge between abstract angle measures and real-world distances. Every engineering, physics, and navigation problem that involves angles uses these ratios — and the special triangles give you exact values without a calculator.
Essential Question
How can the ratios of sides in a right triangle unlock the measurement of any angle or distance — even one you cannot directly measure?
Lesson Overview
In this lesson we define the six trigonometric functions using right triangles, master the special 30-60-90 and 45-45-90 triangles, solve for missing sides and angles, and apply these tools to real-world height and distance problems.
SOH-CAH-TOA — The Six Trig Ratios
For any acute angle θ in a right triangle, the six trigonometric ratios are defined by the relationships between the three sides: the side opposite θ, the side adjacent to θ, and the hypotenuse (the longest side, always opposite the right angle).
Reference — All Six Trig Ratios
Memory trick: SOH-CAH-TOA — Sine=Opp/Hyp, Cosine=Adj/Hyp, Tangent=Opp/Adj
Special Right Triangles
Two special triangles give exact trig values without a calculator. Memorize their side ratios — they appear constantly in precalculus and calculus.
30-60-90 Triangle
Sides: 1 · √3 · 2
sin 30° = 1/2, cos 30° = √3/2
sin 60° = √3/2, cos 60° = 1/2
45-45-90 Triangle
Sides: 1 · 1 · √2
sin 45° = cos 45° = √2/2
tan 45° = 1
Solving Right Triangles
To solve a right triangle means to find all unknown sides and angles. You need at least one side and one acute angle (or two sides). Use the appropriate trig ratio, then solve algebraically. Use inverse trig (arcsin, arccos, arctan) to find unknown angles.
Angles of Elevation and Depression
Real-world problems often involve looking up at an object (angle of elevation) or down at an object (angle of depression). Both are measured from the horizontal. Because of alternate interior angles, the angle of elevation from the ground equals the angle of depression from the top.
Cofunction Identities
The two acute angles in a right triangle always sum to 90° — they are complementary. This means each trig function of one angle equals the cofunction of the other angle.
Key Vocabulary
Sine (sin θ)
The ratio of the side opposite angle θ to the hypotenuse: sin θ = opp/hyp.
Cosine (cos θ)
The ratio of the side adjacent to angle θ to the hypotenuse: cos θ = adj/hyp.
Tangent (tan θ)
The ratio of the side opposite angle θ to the side adjacent: tan θ = opp/adj.
Cosecant (csc θ)
The reciprocal of sine: csc θ = hyp/opp.
Secant (sec θ)
The reciprocal of cosine: sec θ = hyp/adj.
Cotangent (cot θ)
The reciprocal of tangent: cot θ = adj/opp.
Angle of elevation
The angle measured upward from the horizontal to a line of sight.
Angle of depression
The angle measured downward from the horizontal to a line of sight.
Cofunction identity
An identity relating a trig function of θ to its cofunction of (90°−θ), e.g. sin θ = cos(90°−θ).
Inverse trig function
Functions arcsin, arccos, arctan that return an angle when given a ratio.
Worked Examples
A right triangle has legs of length 5 and 12. Find all six trig ratios for the acute angle θ opposite the side of length 5.
First find the hypotenuse: c = √(5² + 12²) = √(25 + 144) = √169 = 13.
Identify the sides relative to θ: opp = 5, adj = 12, hyp = 13.
sin θ = opp/hyp = 5/13
cos θ = adj/hyp = 12/13
tan θ = opp/adj = 5/12
csc θ = hyp/opp = 13/5
sec θ = hyp/adj = 13/12
cot θ = adj/opp = 12/5
Find the exact values of sin 60°, cos 60°, and tan 60° using the 30-60-90 special triangle.
In a 30-60-90 triangle, the sides are in ratio 1 : √3 : 2 (short leg : long leg : hyp).
For the 60° angle: opp = √3, adj = 1, hyp = 2.
sin 60° = opp/hyp = √3/2
cos 60° = adj/hyp = 1/2
tan 60° = opp/adj = √3/1 = √3
Solve the right triangle: θ = 40°, adjacent side = 15. Find the opposite side and hypotenuse.
To find the opposite side, use tangent: tan θ = opp/adj.
tan 40° = opp/15 → opp = 15 · tan 40°
opp = 15 × 0.8391 ≈ 12.59
To find the hypotenuse, use cosine: cos θ = adj/hyp.
cos 40° = 15/hyp → hyp = 15/cos 40°
hyp = 15/0.7660 ≈ 19.58
A person stands 80 ft from the base of a building and looks up at the top with an angle of elevation of 52°. How tall is the building?
Draw a right triangle: the horizontal distance (adj) = 80 ft, the angle of elevation = 52°, and the building height = opp.
Use tangent: tan 52° = opp/adj = height/80.
height = 80 · tan 52°
height = 80 × 1.2799 ≈ 102.4 ft
Find angle θ in a right triangle where the opposite side = 7 and the hypotenuse = 10. Then use a cofunction identity to find the complementary angle's sine.
sin θ = opp/hyp = 7/10 = 0.7
θ = arcsin(0.7) ≈ 44.4°
The complementary angle is 90° − 44.4° = 45.6°.
By the cofunction identity: sin θ = cos(90°−θ), so cos 45.6° = sin 44.4° = 0.7.
Verify: sin(44.4°) ≈ 0.700 ✓
Guided Practice
A right triangle has legs 8 and 15. Find sin θ, cos θ, and tan θ for the angle opposite the side of length 8.
Hint: Find the hypotenuse first using the Pythagorean theorem: c = √(8² + 15²). Then identify opp, adj, hyp relative to θ.
Use the 45-45-90 special triangle to find the exact values of sin 45°, cos 45°, and tan 45°.
Hint: In a 45-45-90 triangle the two legs are equal (both = 1) and the hypotenuse = √2. Label opp, adj, hyp for the 45° angle.
Solve the right triangle: one acute angle is 28°, and the hypotenuse is 20. Find both legs.
Hint: Use sin 28° = opp/20 to find the opposite leg, and cos 28° = adj/20 to find the adjacent leg.
From the top of a 60-ft cliff, a lifeguard spots a swimmer at an angle of depression of 18°. How far is the swimmer from the base of the cliff?
Hint: The angle of depression equals the angle of elevation from the swimmer to the top of the cliff (alternate interior angles). Set up tan 18° = 60/distance.
Use a cofunction identity to rewrite tan 72° in terms of a cotangent, then verify numerically.
Hint: The cofunction identity is tan θ = cot(90°−θ). Substitute θ = 72° and compute both sides with a calculator.
Check Your Understanding
Interactive Practice — 5 Questions
In a right triangle, sin θ = 3/5. What is cos θ?
What is the exact value of sin 45°?
A right triangle has θ = 30° and hypotenuse = 10. What is the side opposite θ?
Which cofunction identity is correct?
A person 50 m from a building looks up at 60°. How tall is the building?
Common Mistakes
Mixing up opposite and adjacent sides when the angle changes.
Opposite and adjacent are always defined relative to the angle θ you are working with — they swap when you switch to the other acute angle.
Using sin θ = adj/hyp instead of opp/hyp.
SOH: Sine = Opposite over Hypotenuse. Cosine = Adjacent over Hypotenuse (CAH). Never swap them.
Forgetting to use inverse trig when solving for an unknown angle.
If you know a ratio and need the angle, apply arcsin, arccos, or arctan. For example, if sin θ = 0.6, then θ = arcsin(0.6) ≈ 36.9°.
Confusing angle of elevation with angle of depression.
Elevation is measured upward from horizontal; depression is measured downward. Both are always positive acute angles.
Math Tips
Memorize the 30-60-90 and 45-45-90 side ratios — they give exact answers and appear on every standardized test.
Quick check: in any right triangle, the hypotenuse is always the longest side and is always opposite the 90° angle.
When setting up an elevation/depression problem, always draw a diagram first and label the horizontal, vertical, and hypotenuse sides clearly.
Cofunction identities only work for complementary angles (summing to 90°). They do NOT apply to supplementary angles.
To remember reciprocals: csc goes with sin (both have an "s"), sec goes with cos (both have a "c"), cot goes with tan (both have a "t").