Key points about exact trigonometric values
A branch of mathematics which explores the relationships between sides and angles in a triangle. can be performed without a scientific calculator for a select number of angles.
The exact trigonometric values that need to be learned are:
| 0° | 30° | 45° | 60° | 90° | |
|---|---|---|---|---|---|
| sinθ | 0 | \( \frac{1}{2}\) | \( \frac{1}{√2}\) or \( \frac{√2}{2}\) | \( \frac{√3}{2}\) | 1 |
| cosθ | 1 | \( \frac{√3}{2}\) | \( \frac{1}{√2}\) or \( \frac{√2}{2}\) | \( \frac{1}{2}\) | 0 |
| tanθ | 0 | \( \frac{1}{√3}\) or \( \frac{√3}{3}\) | 1 | \(√3\) | Undefined |
- Some exact trigonometric values are equivalent. For example, \( \frac{1}{√2}\) = \( \frac{√2}{2}\). The denominator has been rationalised.
To be successful with exact trigonometric values, especially at Higher tier, it is essential to be confident with working with surds.
Geometry with exact trigonometric values
To solve right-angle trigonometry problems without a calculator, the exact trigonometric values must be recalled.
Use the three trigonometric A part-to-part comparison. (or formulae) and the exact trigonometric values to find unknown sides and angles in a right-angled triangle.
The formula used will depend on what information is given in the question.
- The formulae for trigonometry can be remembered using the mnemonic SOHCAHTOA. Try making up your own!
Follow the worked example below
Check your understanding
GCSE exam-style questions
- Work out the value of angle θ using trigonometry.
θ = 45°
Label the sides of the triangle. The hypotenuse (hyp) is the longest side and is opposite the right angle. The opposite side (opp) is the side opposite the given angle, θ. The adjacent side (adj) is the final side next to the given angle.
In this triangle the opposite (opp) and the adjacent (adj) sides are known.
The trigonometric ratio needed must contain the opposite and the adjacent.
The correct formula to use is the tangent ratio, tanθ = opp ÷ adj.
- Substitute the values of opp, adj and θ into the formula to form an equation.
Here the opposite is 8, the adjacent is 8 and the angle should be substituted with θ.
This gives tanθ = 8 ÷ 8.
8 ÷ 8 = 1, so tanθ = 1
- Recall the exact trigonometric value for tanθ = 1.
tan(45) = 1
Hence angle θ = 45°.
- Find the size of length AB using trigonometry.
AB = 3 cm
Label the sides of the triangle. The hypotenuse (hyp) is the longest side and is opposite the right angle. The opposite side (opp) is the side opposite the given angle. The adjacent side (adj) is the final side next to the given angle.
In this triangle the hypotenuse (hyp) is known and the adjacent (adj) is the side to be calculated, AB. The trigonometric ratio needed must contain the adjacent and the hypotenuse. The correct formula to use is the cosine ratio, cosθ = adj ÷ hyp.
Write down the formula cosθ = adj ÷ hyp.
Substitute the values of θ, adj and hyp into the formula to form an equation. Here θ = 60°, the adjacent should be substituted with AB and the hypotenuse is 6.
This gives cos(60) = AB ÷ 6.
- Recall the exact trigonometric value for cos(60) equals ½.
Substitute this into the left-hand side of the equation, to give ½ = AB ÷ 6.
Rearrange the equation to make AB the subject.
Find the value of AB by multiplying both sides of the equation by 6.
This gives ½ × 6 = AB.
This simplifies to AB = 3 cm.
Quiz – Exact trigonometric values
Practise what you've learned about exact trigonometric values with this quiz.
Higher – How to use exact trigonometric values
Exact trigonometric values can be used in calculations.
Many of the exact trigonometric values are written as A number expressed as a square root..
Use the rules for calculating with surds to write answers in their simplest form.
Follow the worked example below
GCSE exam-style questions
- Without using a calculator, work out the value of ⁷⁄₃ × sin(60) × cos(60) × tan(60).
⁷⁄₄ or 1¾.
Recall the values for sin(60), cos(60) and tan(60). sin(60) = √3/2, cos(60) = ½ and tan(60) = √3.
Substitute the values into the expression.
This gives ⁷⁄₃ × √3/2 × ½ × √3.
√3/2 can be written as ½ × √3, so the expression can be rewritten as ⁷⁄₃ × ½ × √3 × ½ × √3.
Multiply the fractions and multiply the surds.
⁷⁄₃ × ½ × ½ = ⁷⁄₁₂ and √3 × √3 = 3.
Therefore the expression becomes ⁷⁄₁₂ × 3.
So ⁷⁄₁₂ × 3 = ²¹⁄₁₂.
This fraction can be simplified by dividing the numerator and denominator by 3, to get the improper fraction ⁷⁄₄, or as a mixed number 1¾.
- Without using a calculator, work out the value of (tan(30) × sin(45) × sin(60))².
⅛
Recall the values for tan(30), sin(45) and sin(60). tan(30) = √3/3, sin(45) = √2/2 and sin(60) = √3/2.
Substitute the values into the expression.
This gives (√3/3 × √2/2 × √3/2)².
- Simplify the bracket by multiplying the numerators and denominators.
√3 × √2 × √3 = √18, and 3 × 2 × 2 = 12.
The bracket is equivalent to √18/12.
- Square the bracket by multiplying √18/12 by itself.
√18/12 × √18/12 = ¹⁸⁄₁₄₄
- Simplify the fraction by dividing the numerator and denominator by 18, to give ⅛.
Higher – Using exact trigonometric values with geometry problems
A multi-step problem may combine Pythagoras’ theorem states the relationship between sides in a right-angled triangle. It states that 𝒂² + 𝒃² = 𝒄², where 𝒄 is the hypotenuse (longest side), and 𝒂 and 𝒃 are the other two sides. with trigonometry.
If the sides of the right-angled triangle are labelled 𝑎, 𝑏 and 𝑐, then Pythagoras' theorem can be written as the A fact, rule, or principle that is expressed in terms of mathematical symbols. The plural of formula is formulae.:
𝑎² + 𝑏² = 𝑐²
Many of the exact trigonometric values are written as surds, so it is common for final answers also to be expressed as a surd.
Follow the worked example below
GCSE exam-style questions
- In the given triangle, what is the exact value of cosθ?
3/√10
Label the sides of the triangle. The hypotenuse (hyp) is the longest side and is opposite the right angle. The opposite side (opp) is the side opposite the given angle. The adjacent side (adj) is the final side next to the given angle.
Write down the formula cosθ = adj/hyp.
Substitute the values of θ, adj and hyp into the formula to form an equation.
Here the adjacent is 3 and the hypotenuse is √10.
This gives cosθ = 3/√10.
In the triangle sinθ = 1/√10 and tanθ = 1/3.
- The length of 𝑄𝑅 equals 𝑥/√3 cm, where 𝑥 is an integer.
Work out the length of 𝑄𝑅.
𝑄𝑅 = 4√3 cm
- Label the sides of the triangle. The hypotenuse (hyp) is the longest side and is opposite the right angle. The opposite side (opp) is the side opposite the given angle. The adjacent side (adj) is the final side next to the given angle.
In this triangle the adjacent (adj) is known and the opposite (opp) is the side to be calculated, QR. The trigonometric ratio needed must contain the opposite and the adjacent. The correct formula to use is the tangent ratio, tanθ = opp/adj.
- Write down the formula tanθ = opp/adj and substitute the values of θ, opp and adj into the formula to form an equation.
Here θ = 30°, the opposite should be substituted with QR and the adjacent is 12.
This gives tan(30) = QR/12.
Recall the exact trigonometric value for tan(30). tan(60) = √3/3 and substitute this into the left-hand side of the equation to give
√3/3 = QR/12.Rearrange the equation to make QR the subject.
Find the value of QR by multiplying both sides of the equation by 12.
This gives 12√3/3 = QR.
- Simplify the surd further by dividing the numerator and denominator by three.
This gives QR = 4√3 cm.
Higher – Quiz – Exact trigonometric values
Practise what you've learned about exact trigonometric values with this quiz for Higher tier.
Now you've revised exact trigonometric values, why not look at solving 2D and 3D problems using Pythagoras' theorem?
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