Non-calculator trigonometry using exact values
Trigonometry is everywhere—from designing buildings to coding 3D video games. In this topic, you'll learn how to work out exact values of sine, cosine, and tangent for special angles like 30°, 45°, and 60°, all without a calculator. These exact values help build a solid foundation for more advanced maths, including A-levels and beyond. Jump to the questions
Practise now
- Use
sqrt(...)for square roots, e.g.sqrt(3) - Use
/for fractions, e.g.1/2orsqrt(3)/2 - Use
*for multiplication, e.g.4*sqrt(2) - Your answers must be exact. No rounded decimals!
Topic guide
What this worksheet practises
This worksheet provides practice on using trigonometric ratios (Sin, Cos, Tan) without a calculator. You are expected to have memorised the "exact values" (written as surds or fractions) for five specific angles: 0°, 30°, 45°, 60°, and 90°.
Key method
There are memory tricks (like the left-hand trick or drawing two specific triangles) to recall these values, but you must know them to answer the questions.
- Identify the correct ratio using SOH CAH TOA, just like normal trigonometry.
- Set up your equation (e.g. sin(30°) = x / 12).
- Replace the trigonometric part (e.g. sin(30°)) with its memorised exact value fraction.
- Solve the resulting equation using standard algebra or fraction multiplication.
Worked example
A right-angled triangle has an angle of 60°. The hypotenuse is 10cm. Find the exact length of the opposite side.
Step 1: We know the Hypotenuse (H) and want the Opposite (O). This means using Sin (SOH).
Step 2: Set up the equation.
sin(60°) = O / 10
Step 3: Rearrange to solve for O.
O = 10 × sin(60°)
Step 4: Recall the exact value for sin(60°), which is √3/2. Substitute this into the equation.
O = 10 × (√3 / 2)
Step 5: Multiply. (10 ÷ 2 is 5).
O = 5√3 cm.
Common mistakes to avoid
The biggest hurdle is simply misremembering the table of values. A very common confusion is swapping the values for sin(30) and cos(30). Remember that sin(30) is the simple fraction (1/2), while cos(30) is the surd (√3/2).
Things to remember
The value for tan(45°) is exactly 1. This is because a right-angled triangle with a 45° angle is isosceles, meaning its opposite and adjacent sides are exactly the same length. Any number divided by itself is 1.