Exact Trigonometric Values
While your calculator is great for most trig problems, for certain special angles ($0°, 30°, 45°, 60°, 90°$), you are expected to know the exact answers without a calculator. These precise values — often involving fractions and surds — are crucial for accuracy in engineering, design, and higher-level maths.
Deriving the Exact Values
Instead of just memorising, you can derive most of these values from two special triangles.
The 45-45-90 Triangle
Start with a square of side length 1 and cut it in half diagonally. This creates a right-angled isosceles triangle.
- $\sin 45° = \dfrac{1}{\sqrt{2}} = \dfrac{\sqrt{2}}{2}$
- $\cos 45° = \dfrac{1}{\sqrt{2}} = \dfrac{\sqrt{2}}{2}$
- $\tan 45° = \dfrac{1}{1} = 1$
The 30-60-90 Triangle
Start with an equilateral triangle with side length 2 and cut it in half vertically. The dashed line shows the cut (height = √3).
- $\sin 30° = \dfrac{1}{2}$, $\cos 30° = \dfrac{\sqrt{3}}{2}$
- $\sin 60° = \dfrac{\sqrt{3}}{2}$, $\cos 60° = \dfrac{1}{2}$
- $\tan 60° = \sqrt{3}$
The Exact Values Table
| Angle ($\theta$) | $0°$ | $30°$ | $45°$ | $60°$ | $90°$ |
|---|---|---|---|---|---|
| $\sin\theta$ | $0$ | $\dfrac{1}{2}$ | $\dfrac{\sqrt{2}}{2}$ | $\dfrac{\sqrt{3}}{2}$ | $1$ |
| $\cos\theta$ | $1$ | $\dfrac{\sqrt{3}}{2}$ | $\dfrac{\sqrt{2}}{2}$ | $\dfrac{1}{2}$ | $0$ |
| $\tan\theta$ | $0$ | $\dfrac{\sqrt{3}}{3}$ | $1$ | $\sqrt{3}$ | Undefined |
Worked Example
Find the exact length of side $x$.
- Label sides: From the $60°$ angle, 5 cm is the Adjacent and $x$ is the Hypotenuse.
- Choose ratio: We have A and H, so we use CAH (Cosine).
$\cos(60°) = \dfrac{\text{Adjacent}}{\text{Hypotenuse}} = \dfrac{5}{x}$. - Substitute exact value: $\cos(60°) = \dfrac{1}{2}$.
$\dfrac{1}{2} = \dfrac{5}{x}$. - Solve: $x = 5 \times 2 = 10$ cm.
Answer: $x = 10$ cm.
Tutor Insights
🤔 Common Misunderstandings
- “Can’t I just use a calculator?” This is a non-calculator topic — exam questions are specifically designed to test your knowledge of exact values.
- Mixing up values: Accidentally using $\sin(60°)$ instead of $\cos(60°)$.
- Surd confusion: Struggling to simplify expressions involving $\sqrt{2}$ or $\sqrt{3}$.
📝 Common Exam Mistakes
- Providing a decimal answer when an exact value is required.
- Errors with fraction arithmetic when combining exact values.
- Forgetting that $\tan(90°)$ is undefined.
- Not showing the substitution step — marks are awarded for demonstrating you know the correct exact value.
Practice Questions
- Calculate the exact value of $\sin(30°) + \cos(60°)$.
- Find the exact value of $6\tan(45°) + 2\cos(30°)$.
- Show that $2\cos^2(30°) – \sin(90°) = \dfrac{1}{2}$. (Note: $\cos^2\theta$ means $(\cos\theta)^2$.)
Show Answers
- Working: $\sin(30°) = \frac{1}{2}$, $\cos(60°) = \frac{1}{2}$. $\frac{1}{2} + \frac{1}{2} = 1$.
Answer: 1. - Working: $\tan(45°) = 1$, $\cos(30°) = \frac{\sqrt{3}}{2}$. $6(1) + 2\!\left(\frac{\sqrt{3}}{2}\right) = 6 + \sqrt{3}$.
Answer: $6 + \sqrt{3}$. - Working: $2\!\left(\frac{\sqrt{3}}{2}\right)^{\!2} – 1 = 2\!\left(\frac{3}{4}\right) – 1 = \frac{3}{2} – 1 = \frac{1}{2}$. ∎
Answer: Proof complete.
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