Trigonometry from a Triangle: sin, cos and tan
A ladder, a wall, and one angle
A ladder rests against a vertical wall on level ground. The wall, ground and ladder form a right triangle. Trigonometry connects an acute angle of that triangle with ratios of its side lengths.
In this lesson you will identify the sides correctly, choose sin, cos or tan, and find a missing length. You need only fractions and the Pythagorean theorem. All angles here are in degrees and the chosen angle is acute: 0° < θ < 90°.
1. Choose the angle before naming the sides
In triangle ABC, angle C is 90°. We first choose θ at B.
- Hypotenuse (H): AB, opposite the 90° angle. It is the longest side.
- Opposite (O): AC, directly across from θ. It does not touch θ.
- Adjacent leg (A): BC, the side touching θ that is not the hypotenuse.
Here O, A and H are shorthand for side lengths. The shorthand A for “adjacent” is not the vertex A of the diagram.
The words “opposite” and “adjacent” depend on your chosen angle. The hypotenuse does not change.
At θ: opposite = 3, adjacent = 4. At α: opposite = 4, adjacent = 3. Do not label a side “opposite” without first identifying the angle.
2. Three ratios, not three new lengths
For the chosen acute angle:
- sin θ = opposite / hypotenuse = O/H
- cos θ = adjacent leg / hypotenuse = A/H
- tan θ = opposite / adjacent leg = O/A
A memory aid is SOH–CAH–TOA: Sine = Opposite/Hypotenuse; Cosine = Adjacent/Hypotenuse; Tangent = Opposite/Adjacent.
For θ in the 3–4–5 triangle: sin θ = 3/5, cos θ = 4/5, tan θ = 3/4. For α: sin α = 4/5, cos α = 3/5, tan α = 4/3.
A ratio has no length unit when both lengths use the same unit. Convert units before dividing. For an acute angle, sine and cosine are between 0 and 1, but tangent can exceed 1.
3. Why the triangle's size does not matter
Enlarge the 3–4–5 triangle to a similar 6–8–10 triangle. Its corresponding angles stay equal, and sin θ is still 6/10 = 3/5. Cosine is 8/10 = 4/5 and tangent is 6/8 = 3/4.
Same acute angle + a right angle ⇒ similar triangles ⇒ the same side ratios. The ratios describe the angle, not how large you draw the triangle.
Worked example 1: how high does the ladder reach?
A 10 m ladder rests against a vertical wall on level ground. The acute angle θ between the ladder and the ground satisfies sin θ = 3/5. How high above the ground does the top reach?
Try choosing the ratio before revealing the solution.
- The ladder is the hypotenuse: H = 10 m.
- The height up the wall is opposite θ. Call it h.
- Use sin θ = opposite/hypotenuse: 3/5 = h/10.
- Multiply by 10: h = 10 × 3/5 = 6 m.
- Check: 6 m is shorter than the 10 m ladder. The ground distance is √(10² − 6²) = √64 = 8 m, giving a 6–8–10 right triangle.
Answer: 6 m. Using cos θ would connect the ground distance with the ladder, not the height.
Worked example 2: given cosine, find tangent
For an acute angle θ in a right triangle, cos θ = 12/13. Find tan θ.
- Cosine compares adjacent leg with hypotenuse: A/H = 12/13.
- Represent the lengths as A = 12k and H = 13k, where k > 0. They need not literally be 12 and 13 metres.
- By Pythagoras, O² = H² − A² = (13k)² − (12k)² = 25k².
- A length is positive, so O = 5k.
- Therefore tan θ = O/A = 5k/(12k) = 5/12.
Answer: 5/12. The ratio 12/13 is cosine, not tangent. The missing side comes from squares: do not subtract 13 − 12 to find it.
A 15-second solving checklist
- Mark the right angle and the chosen acute angle.
- Find the hypotenuse, then the opposite and adjacent legs.
- Choose the ratio containing the known and required lengths.
- Substitute, solve, and check the units and size.
Watch out: sin θ does not mean “sin × θ.” The name sin refers to the sine function; sin θ is its value at θ. The diagram's orientation does not change the ratios, and “adjacent” here excludes the hypotenuse.
Eight original practice questions
Each question uses a right triangle and an acute chosen angle. No calculator is needed.
- Triangle ABC is right-angled at C. For the angle at B, which side is opposite?
Show answer
Correct answer: C
AC is across from B. AB is the hypotenuse and BC is the adjacent leg.
Hint: Find the side that does not touch B.
- Relative to θ, the opposite leg is 5 cm and the hypotenuse is 13 cm. What is sin θ?
Show answer
Correct answer: D
sin θ = opposite/hypotenuse = 5/13. The adjacent leg is not needed.
Hint: Use SOH.
- Relative to θ, the adjacent leg is 8 m and the hypotenuse is 10 m. What is cos θ?
Show answer
Correct answer: B
cos θ = adjacent/hypotenuse = 8/10 = 4/5.
Hint: Use CAH and simplify the fraction.
- Relative to θ, the opposite leg is 12 cm and the adjacent leg is 5 cm. What is tan θ?
Show answer
Correct answer: C
tan θ = opposite/adjacent = 12/5. Tangent can be greater than 1.
Hint: The denominator is the adjacent leg, not the hypotenuse.
- A right triangle has hypotenuse 15 cm and sin θ = 3/5. Find the opposite leg.
Show answer
Correct answer: A
opposite = hypotenuse × sin θ = 15 × 3/5 = 9 cm.
Hint: Write opposite/15 = 3/5.
- For an acute angle θ, cos θ = 8/17. Find tan θ.
Show answer
Correct answer: D
Take adjacent = 8k and hypotenuse = 17k, with k > 0. Opposite = √(289k² − 64k²) = 15k. Thus tan θ = 15k/(8k) = 15/8.
Hint: Find the third side with Pythagoras before taking the ratio.
- Triangle ABC is right-angled at C, with AC = 3, BC = 4 and AB = 5. If the chosen angle is moved from B to A, what is its sine?
Show answer
Correct answer: B
For angle A, the opposite side is BC = 4 and the hypotenuse is AB = 5. Hence sin A = 4/5.
Hint: Rename opposite and adjacent after changing the angle.
- Relative to θ, the opposite leg is 60 cm and the adjacent leg is 0.8 m. What is tan θ?
Show answer
Correct answer: C
Convert 0.8 m to 80 cm. Then tan θ = 60/80 = 3/4. Dividing 60 by 0.8 without converting mixes units.
Hint: Use the same unit for both lengths.
Take away one idea
Choose the angle → name the sides → choose the ratio. That order matters more than memorising a page of formulas.
Sign in to keep your progress. Sign in