Trigonometry Formulas: Sin, Cos, Tan and Special Angles Table for Grade 10

The trigonometry formulas connect angles and sides in a right triangle through three fundamental ratios: sin θ = opposite ÷ hypotenuse, cos θ = adjacent ÷ hypotenuse, tan θ = opposite ÷ adjacent. This material is mandatory in Grade 10 senior high school in Indonesia and frequently appears in UTBK/SNBT and physics problems.

This article gathers the basic trigonometry formulas, the special-angle table that must be memorised, fully worked examples with step-by-step solutions, SOH-CAH-TOA memorisation tips, and how this material connects to UTBK 2026 preparation.

Trigonometry and its origins

Trigonometry is the branch of mathematics that studies the relationships between angles and sides of right triangles. The word comes from Greek: trigonon (triangle) and metron (measurement). Hipparchus of Ancient Greece first developed a trigonometry table around 140 BC.

Trigonometry is used daily by architects, engineers, and navigators - from calculating roof slopes to determining the position of a ship at sea. The material is studied in Grade 10 senior high school as part of the required mathematics curriculum.

The sin, cos, and tan formulas

Three primary trigonometric ratios for angle θ in a right triangle:

In words: sin θ = opposite ÷ hypotenuse, cos θ = adjacent ÷ hypotenuse, tan θ = opposite ÷ adjacent.

💡 Memorisation Tip: Use the SOH-CAH-TOA mnemonic: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.

How to calculate trigonometry: worked examples

Example 1: Computing sin, cos, and tan values

A builder examines a house's roof frame. He measures the rafter (roof beam) length = 5 m, vertical wall height = 3 m, and horizontal distance from wall to rafter end = 4 m. Calculate sin θ, cos θ, and tan θ for the angle θ between the rafter and the horizontal.

Step 1: Identify all sides

• Opposite (vertical) = 3 m
• Adjacent (horizontal) = 4 m
• Hypotenuse (rafter) = 5 m

Step 2: Plug into the formulas

sin θ = 3/5 = 0.6 | cos θ = 4/5 = 0.8 | tan θ = 3/4 = 0.75

✅ Answer: sin θ = 0.6 | cos θ = 0.8 | tan θ = 0.75

Example 2: Finding an unknown side

A ladder leans against a wall at a 30° angle to the floor. The ladder is 10 m long. How high up the wall does the ladder reach?

Step 1: Given: angle θ = 30°, hypotenuse = 10 m. Find: opposite (wall height).

Step 2: Use the sin formula (we know the hypotenuse, find the opposite)

Step 3: Solve (sin 30° = 0.5)

✅ Answer: The wall height reached is 5 m

Example 3: Common mistake - swapping sin and cos

A student wants to find the adjacent side with θ = 30° and hypotenuse = 10 m. They use the sin formula. Is this correct?

❌ Common Mistake: The sin formula gives the opposite side, not the adjacent! The result is wrong.

✅ Correct (use cos for adjacent):

adjacent = cos 30° × 10 = (√3/2) × 10 ≈ 8.66 m

✅ Answer: adjacent = 5√3 ≈ 8.66 m

Key: Remember CAH - Cosine = Adjacent / Hypotenuse.

Special-angle trigonometry table

Memorise this table to solve problems without a calculator.

Tips and tricks for memorising trigonometry

1. Use SOH-CAH-TOA

The most popular mnemonic to remember three formulas at once: Sin=Opposite/Hyp · Cos=Adjacent/Hyp · Tan=Opposite/Adjacent.

2. The pattern of sin and cos values in the table

For sin, the order from 0° to 90° is: 0, 1/2, √2/2, √3/2, 1. For cos, the order is exactly reversed: 1, √3/2, √2/2, 1/2, 0. Memorise one row, and the other follows automatically.

3. Never swap opposite and adjacent

The most common mistake: using sin for adjacent (should be cos) or vice versa. Always identify the reference angle θ first, then identify opposite and adjacent relative to that angle.

Trigonometry in UTBK and senior-school prep

For Grade 11 and 12 students preparing for UTBK-SNBT (Indonesia's university entrance test), trigonometry is a required topic in the Mathematical Reasoning subtest. UTBK trigonometry problems usually combine several concepts at once: sin/cos/tan ratios, basic trigonometric identities, and word problems like tower heights or distance from observation.

An effective approach: master the basic formulas from this article first, then practice 5-10 past UTBK problems weekly. The special-angle table must be memorised until it can be reproduced without thinking.

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Summary

1. Trigonometry studies the relationship between angles and sides in a right triangle - required material in Grade 10.
2. Three main formulas: sin θ = opposite/hypotenuse; cos θ = adjacent/hypotenuse; tan θ = opposite/adjacent.
3. Use SOH-CAH-TOA to remember all three formulas at once without confusion.
4. The special-angle trigonometry values (0°, 30°, 45°, 60°, 90°) must be memorised - they appear constantly in UTBK problems and school exams.
5. Always identify the reference angle before opposite and adjacent so they do not get swapped.

Trigonometry practice problems

⭐ Problem 1 (Easy): A right triangle with angle α has opposite = 5 and hypotenuse = 13. Calculate sin α, cos α, and tan α.

⭐⭐ Problem 2 (Medium): Given cos β = 0.8 and hypotenuse = 20 cm. What is the adjacent length?

⭐⭐ Problem 3 (Medium): A 30 m tower. An observer stands 30 m from the tower's base. What is the tan of the elevation angle to the tower top, and what is the angle?

⭐⭐⭐ Problem 4 (Hard): Calculate: sin 60° × cos 30° + cos 60° × sin 30°.

⭐⭐⭐ Problem 5 (Hard / UTBK-style): Given sin α = 3/5 and α is acute. Determine cos α and tan α without using a calculator.

Answer Key:

1. Adjacent = √(13²−5²) = √144 = 12 → sin α = 5/13 ≈ 0.385 | cos α = 12/13 ≈ 0.923 | tan α = 5/12 ≈ 0.417. Use the Pythagorean theorem to find the unknown side before computing ratios.
2. Adjacent = cos β × hypotenuse = 0.8 × 20 = 16 cm. From the cos formula: adjacent = cos β × hypotenuse.
3. tan = opposite/adjacent = 30/30 = 1 → angle = 45°. tan 45° = 1 because the opposite equals the adjacent at 45°.
4. (√3/2 × √3/2) + (1/2 × 1/2) = 3/4 + 1/4 = 1. Angle sum formula: sin(A+B) = sin A cos B + cos A sin B. With A=60°, B=30°: sin 90° = 1.
5. Use the identity sin²α + cos²α = 1: cos²α = 1 − (3/5)² = 1 − 9/25 = 16/25 → cos α = 4/5 (positive since α is acute). Then tan α = sin α / cos α = (3/5) / (4/5) = 3/4. The Pythagorean trig identity is the formula most frequently appearing in UTBK.