Use exact ratios and preserve surds rather than calculator decimals.
Extended · Syllabus E6.3
Notes and key methods
The exact sine values at 0°, 30°, 45°, 60° and 90° are \(\displaystyle 0, \frac{1}{2}, \frac{\sqrt{2}}{2}, \frac{\sqrt{3}}{2}\) and 1. Cosine reverses that list. Tangent is sine divided by cosine and is undefined at 90°.
Special right triangles establish these values: a 45° triangle has side ratio \(\displaystyle 1:1:\sqrt{2}\) and half an equilateral triangle has ratio \(\displaystyle 1:\sqrt{3}:2\). Retain exact surds in non-calculator work and use the acute-angle side ratio appropriate to the question.
Check the conditions
Use the stated units and precision.
Show the method as well as the final answer.
Three worked examples
Example 1 · Exact special-angle trigonometry
Use exact trigonometric values. All angles are measured in degrees.
Choose the exact value of sin 45°.
\(\displaystyle \frac{\sqrt{2}}{2} + 1\)
\(\displaystyle \frac{\sqrt{2}}{2} + 2\)
\(\displaystyle \frac{\sqrt{2}}{2} + 3\)
\(\displaystyle \frac{\sqrt{2}}{2}\)
Choose the exact value of cos 45°.
\(\displaystyle \frac{\sqrt{2}}{2} + 2\)
\(\displaystyle \frac{\sqrt{2}}{2} + 3\)
\(\displaystyle \frac{\sqrt{2}}{2}\)
\(\displaystyle \frac{\sqrt{2}}{2} + 1\)
Choose the exact value of tan 45°.
\(4\)
\(1\)
\(2\)
\(3\)
A right triangle has hypotenuse 12 cm and acute angle 45°. Choose its exact opposite length in cm.
\(6 \sqrt{2}\)
\(1 + 6 \sqrt{2}\)
\(2 + 6 \sqrt{2}\)
\(3 + 6 \sqrt{2}\)
Show the reasoning
(a)
Use a special right-triangle ratio and retain its exact value.
The special-angle ratio gives \(\displaystyle \frac{\sqrt{2}}{2}\).
Answer: \(\displaystyle \frac{\sqrt{2}}{2}\).
(b)
Use a special right-triangle ratio and retain its exact value.
The special-angle ratio gives \(\displaystyle \frac{\sqrt{2}}{2}\).
Answer: \(\displaystyle \frac{\sqrt{2}}{2}\).
(c)
Use a special right-triangle ratio and retain its exact value.
The special-angle ratio gives \(1\).
Answer: \(1\).
(d)
Use a special right-triangle ratio and retain its exact value.
The special-angle ratio gives \(6 \sqrt{2}\).
Answer: \(6 \sqrt{2}\).
Example 2 · Exact special-angle trigonometry
Use exact trigonometric values. All angles are measured in degrees.
Choose the exact value of sin 60°.
\(\displaystyle \frac{\sqrt{3}}{2} + 2\)
\(\displaystyle \frac{\sqrt{3}}{2} + 3\)
\(\displaystyle \frac{\sqrt{3}}{2}\)
\(\displaystyle \frac{\sqrt{3}}{2} + 1\)
Choose the exact value of cos 60°.
\(\displaystyle \frac{7}{2}\)
\(\displaystyle \frac{1}{2}\)
\(\displaystyle \frac{3}{2}\)
\(\displaystyle \frac{5}{2}\)
Choose the exact value of tan 60°.
\(\sqrt{3}\)
\(1 + \sqrt{3}\)
\(\sqrt{3} + 2\)
\(\sqrt{3} + 3\)
A right triangle has hypotenuse 14 cm and acute angle 60°. Choose its exact opposite length in cm.
\(1 + 7 \sqrt{3}\)
\(2 + 7 \sqrt{3}\)
\(3 + 7 \sqrt{3}\)
\(7 \sqrt{3}\)
Show the reasoning
(a)
Use a special right-triangle ratio and retain its exact value.
The special-angle ratio gives \(\displaystyle \frac{\sqrt{3}}{2}\).
Answer: \(\displaystyle \frac{\sqrt{3}}{2}\).
(b)
Use a special right-triangle ratio and retain its exact value.
The special-angle ratio gives \(\displaystyle \frac{1}{2}\).
Answer: \(\displaystyle \frac{1}{2}\).
(c)
Use a special right-triangle ratio and retain its exact value.
The special-angle ratio gives \(\sqrt{3}\).
Answer: \(\sqrt{3}\).
(d)
Use a special right-triangle ratio and retain its exact value.
The special-angle ratio gives \(7 \sqrt{3}\).
Answer: \(7 \sqrt{3}\).
Example 3 · Exact special-angle trigonometry
Use exact trigonometric values. All angles are measured in degrees.
Choose the exact value of sin 30°.
\(\displaystyle \frac{7}{2}\)
\(\displaystyle \frac{1}{2}\)
\(\displaystyle \frac{3}{2}\)
\(\displaystyle \frac{5}{2}\)
Choose the exact value of cos 30°.
\(\displaystyle \frac{\sqrt{3}}{2}\)
\(\displaystyle \frac{\sqrt{3}}{2} + 1\)
\(\displaystyle \frac{\sqrt{3}}{2} + 2\)
\(\displaystyle \frac{\sqrt{3}}{2} + 3\)
Choose the exact value of tan 30°.
\(\displaystyle \frac{\sqrt{3}}{3} + 1\)
\(\displaystyle \frac{\sqrt{3}}{3} + 2\)
\(\displaystyle \frac{\sqrt{3}}{3} + 3\)
\(\displaystyle \frac{\sqrt{3}}{3}\)
A right triangle has hypotenuse 16 cm and acute angle 30°. Choose its exact opposite length in cm.
\(10\)
\(11\)
\(8\)
\(9\)
Show the reasoning
(a)
Use a special right-triangle ratio and retain its exact value.
The special-angle ratio gives \(\displaystyle \frac{1}{2}\).
Answer: \(\displaystyle \frac{1}{2}\).
(b)
Use a special right-triangle ratio and retain its exact value.
The special-angle ratio gives \(\displaystyle \frac{\sqrt{3}}{2}\).
Answer: \(\displaystyle \frac{\sqrt{3}}{2}\).
(c)
Use a special right-triangle ratio and retain its exact value.
The special-angle ratio gives \(\displaystyle \frac{\sqrt{3}}{3}\).
Answer: \(\displaystyle \frac{\sqrt{3}}{3}\).
(d)
Use a special right-triangle ratio and retain its exact value.
The special-angle ratio gives \(8\).
Answer: \(8\).
Common mistakes
Keep intermediate values exact until the final step.