Trigonometry
Grade 7-8Exact Trig Values
A handful of angles give trig values that can be written exactly, as fractions and surds, instead of rounded decimals, and exam questions that say "give your answer as a surd" or "without a calculator" are testing whether these are memorised. This lesson covers the exact values for sine, cosine and tangent at key angles, using them to find exact side lengths and angles, and combining them in calculations.
What you need to know
- You must learn sin, cos and tan of 0, 30, 45, 60 and 90 degrees by heart, with no calculator.
- sin 30 is one half, cos 60 is one half, sin 45 equals cos 45 equals 1 over root 2, and tan 45 is 1.
- sin 60 and cos 30 are both root 3 over 2; tan 60 is root 3 and tan 30 is 1 over root 3.
- tan 90 is undefined.
Written by Asad, co-founder of Teachably
Video walkthrough coming soon
The written lesson below covers everything you need in the meantime.
The exact trig values table
The angles 0°, 30°, 45°, 60° and 90° all have trig values that can be written exactly, without rounding. These need to be memorised, since Higher tier questions expect them to be recalled directly rather than found with a calculator.
sin 30° = 1/2, sin 45° = √2/2, sin 60° = √3/2. cos 30° = √3/2, cos 45° = √2/2, cos 60° = 1/2. tan 30° = √3/3, tan 45° = 1, tan 60° = √3.
Using exact values to find a side length
When a right-angled triangle question involves one of the key angles and asks for an exact answer, substitute the exact trig value into SOHCAHTOA instead of a calculator value, and simplify the resulting surd.
To find x, opposite a 45° angle with a hypotenuse of 8 cm: sin 45° = x ÷ 8, so x = 8 × √2/2, which simplifies to 4√2 cm.
Working backwards from an exact value to an angle
If a question gives an exact trig value and asks for the angle, match it against the memorised table rather than using a calculator.
Given that sin x = √3/2, recognise this as the exact value for sin 60°, so x = 60°.
Combining exact values in a calculation
When an expression combines several exact trig values, substitute each one from the table, then simplify the resulting fractions and surds fully, without switching to decimals at any point.
2 sin 60° - tan 45° = 2 × √3/2 - 1 = √3 - 1.
Worked examples
Three exam-style questions, fully solved
Write down the exact value of cos 60°.
Easy- 1.Recall the exact value from the table
Answer: 1/2
The diagram shows a right-angled triangle with a hypotenuse of 8 cm and a 45° angle. Find the exact length of side x, opposite the 45° angle. Give your answer as a surd in its simplest form.
Medium- 1.Set up the equation using sine: sin 45° = x ÷ 8
- 2.Substitute the exact value: x = 8 × √2/2
- 3.Simplify the surd
Answer: 4√2 cm
Work out the exact value of 3 tan 60° - 2 tan 30°. Give your answer as a surd in its simplest form.
Hard- 1.Substitute the exact values: tan 60° = √3, and tan 30° = √3/3
- 2.Substitute into the expression: 3√3 - 2 × √3/3
- 3.Simplify: 3√3 - 2√3/3 = 9√3/3 - 2√3/3
Answer: 7√3/3
Practice
6 questions, marked instantly
Type an answer and check it. The worked solution appears once you have had a go. No login needed, and your progress saves in this browser.
Practice
Now try these yourself.
Type your answer and check it. The worked solution appears once you have had a go.
Write down the exact value of sin 30°.
Write down the exact value of cos 60°.
Write down the exact value of tan 45°.
Write down the exact value of tan 0°.
Write down the exact value of cos 90°.
Write down the exact value of sin 60°.
Avoid these
The mistakes students make most often
Reaching for a calculator or a rounded decimal instead of recalling the exact fraction or surd value.
Misremembering which exact value belongs to which angle, for example mixing up sin 30° and cos 30°.
Not simplifying a surd fully in the final answer, for example leaving 8 × √2/2 instead of simplifying to 4√2.
When working backwards from an exact value to an angle, matching it to the wrong angle in the table.
Making an arithmetic slip or losing a negative sign when combining several exact trig values in one expression.
FAQ
Questions students and parents ask
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