What is Trigonometry?
Trigonometry studies the relationships between the angles and side lengths of triangles, especially right triangles. It underlies navigation, engineering, physics and computer graphics through three core ratios: sine, cosine and tangent.
Trigonometry relates a right triangle's angles to its side ratios: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent — remembered as SOH-CAH-TOA.
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Step-by-step worked examples
A right triangle has hypotenuse 10 and angle θ = 30°. Find the opposite side.
sin θ = opp/hyp opp = hyp × sin(30°) = 10 × 0.5 = 5
A ramp makes a 20° angle with the ground and is 12 m long (hypotenuse). How high does it rise?
opp = hyp × sin(20°) opp = 12 × 0.342 ≈ 4.10 m
A right triangle has adjacent side 8 and angle θ = 45°. Find the hypotenuse.
cos θ = adj/hyp hyp = adj/cos(45°) = 8/0.707 ≈ 11.31
Flashcards
Quick quiz
Q1.In a right triangle, sin θ equals:
Q2.Hypotenuse = 20, θ = 30°. Opposite side?
Q3.tan θ is defined as:
Q4.What is cos(60°) approximately?
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Common mistakes
Mixing up which side is opposite vs adjacent. — Correct: Opposite/adjacent are defined relative to the chosen angle θ, not fixed positions.
Using degrees when a calculator is set to radians (or vice versa). — Correct: Always check your calculator's angle mode before evaluating sin/cos/tan.
Assuming SOH-CAH-TOA works on any triangle. — Correct: These ratios only apply directly to right triangles; use Law of Sines/Cosines otherwise.
Forgetting the hypotenuse is always the longest side. — Correct: No matter the angle, the hypotenuse (opposite the 90° angle) is always longest.
FAQ
What is trigonometry?
The branch of math studying relationships between triangle angles and side lengths, centered on sin, cos and tan.
What is the trigonometry formula?
sin θ = opp/hyp, cos θ = adj/hyp, tan θ = opp/adj — SOH-CAH-TOA.
How do you calculate trigonometry ratios?
Identify the right triangle's angle, then divide the relevant sides: opposite, adjacent, hypotenuse.
What are examples of trigonometry?
Finding a building's height from its shadow angle, or a ramp's rise given its length and incline angle.