Right triangle trigonometry and the unit circle

IB Mathematics Analysis & Approaches SL· 25 min read

1. Trigonometric Ratios in Right Triangles★☆☆☆☆⏱ 15 min

✓ Calculator OK

📘 Definition

Trigonometric Ratios

$sin θ$ , $cos θ$ , $tan θ$

For an acute angle $θ$ in a right-angled triangle, each ratio is the quotient of two sides relative to $θ$ : $sin θ = \frac{opposite}{hypotenuse}$ , $cos θ = \frac{adjacent}{hypotenuse}$ , $tan θ = \frac{opposite}{adjacent}$

Example:

For a 30° angle with opposite side 2 and hypotenuse 4, $sin 30° = \frac{2}{4} = 0.5$

📐 Worked Example

A right triangle has hypotenuse 10 cm and an angle of 35°. Find the length of the side opposite the 35° angle, correct to 1 decimal place.

1
We know the hypotenuse and need the opposite side, so we use the sine ratio:
2
$\sin 35^\circ = \frac{\text{opposite}}{10}$
3
Rearrange to isolate the unknown opposite side:
4
$\text{opposite} = 10 \times \sin 35^\circ$
5
Calculate with a calculator (ensure it is set to degree mode):
6
$\text{opposite} \approx 10 \times 0.5736 = 5.7 \text{ cm}$

2. Special Right Triangles and Exact Values★★☆☆☆⏱ 20 min

🚫 No Calculator

The 30-60-90 and 45-45-90 special right triangles have simple whole-number side ratios that produce exact trigonometric values. These values are almost always required in non-calculator Paper 1 exam questions.

📘 Definition

Special Right Triangles

Right triangles with angles that produce simple, memorable exact trigonometric values, with side ratios of $1 : 1 : 2$ for 45-45-90 and $1 : 3 : 2$ for 30-60-90.

📐 Worked Example

Find the exact value of $tan 60°$ .

1
Recall the 30-60-90 triangle side ratios: opposite 30° = 1, opposite 60° = $3$ , hypotenuse = 2.
2
For an angle of 60°, the opposite side is $3$ and the adjacent side is 1.
3
$\tan 60^\circ = \frac{\text{opposite}}{\text{adjacent}} = \frac{\sqrt{3}}{1} = \sqrt{3}$

✓ Quick check

Test your knowledge of exact values

What is the exact value of $sin 45°$ ?
- $\frac{1}{2}$
- $\frac{2}{2}$
- $\frac{3}{2}$
- $1$
Reveal answer
$\frac{\sqrt{2}}{2}$ —
Correct! For 45-45-90, opposite = 1, hypotenuse = $2$ , so $sin 45° = \frac{1}{2} = \frac{2}{2}$ after rationalizing.

3. Unit Circle Definition of Trigonometric Functions★★☆☆☆⏱ 20 min

✓ Calculator OK

Right triangle trigonometry only works for acute angles between 0° and 90°. To define trigonometric functions for any angle, positive or negative, we use the unit circle.

📘 Definition

Unit Circle Definition

For any angle $θ$ measured counterclockwise from the positive x-axis, the coordinates of the point where the terminal side of $θ$ intersects the unit circle are $(cos θ, sin θ)$ .

x^2 + y^2 = 1, \quad \cos\theta = x, \quad \sin\theta = y, \quad \tan\theta = \frac{y}{x} = \frac{\sin\theta}{\cos\theta}

📐 Worked Example

Find the value of $cos 180°$ using the unit circle.

1
A 180° angle lies along the negative x-axis. Its intersection point with the unit circle is $(- 1, 0)$ .
2
By definition, the x-coordinate of the point equals $cos θ$ .
3
$\cos 180^\circ = -1$

4. Reference Angles for Exact Values★★★☆☆⏱ 25 min

🚫 No Calculator

To find the exact value of a trigonometric function for any angle, we find the reference angle (the acute angle between the terminal side and the x-axis), get the magnitude from the special triangle, then apply the correct sign based on the quadrant.

📐 Worked Example

Find the exact value of $sin 150°$ .

1
150° is in Quadrant 2, where sine is positive per the ASTC rule.
2
Calculate the reference angle for Quadrant 2: $180° - 150° = 30°$ .
3
We know $sin 30° = \frac{1}{2}$ , so the magnitude is $\frac{1}{2}$ with a positive sign.
4
$\sin 150^\circ = +\frac{1}{2}$

5. Common Pitfalls

Wrong move:

Forgetting to check that your calculator is in degree mode

Why:

IB exams almost always use degrees for trigonometry, and radian mode will give incorrect results

Correct move:

Always confirm your calculator's mode before starting any trig calculation

Wrong move:

Mixing up opposite and adjacent sides relative to the working angle

Why:

Labels are relative to the angle you are using, not fixed to the triangle

Correct move:

Always re-label sides for the specific angle you are working with

Wrong move:

Forgetting to add the negative sign for values outside Quadrant 1

Why:

Students often remember the magnitude from the reference angle but ignore the sign rule

Correct move:

Always check the quadrant and apply the ASTC rule before writing your final answer

Wrong move:

Claiming $tan 90° = 0$

Why:

$tan θ = \frac{s i n θ}{c o s θ}$ , and $cos 90° = 0$ , so division by zero is undefined

Correct move:

Recognize that tangent is undefined at 90°, 270°, and all coterminal angles

6. Quick Reference Cheatsheet

Angle (°)	$sin θ$	$cos θ$	$tan θ$
0	0	1	0
30	$\frac{1}{2}$	$\frac{3}{2}$	$\frac{1}{3}$
45	$\frac{2}{2}$	$\frac{2}{2}$	1
60	$\frac{3}{2}$	$\frac{1}{2}$	$3$
90	1	0	Undefined
180	0	-1	0
270	-1	0	Undefined
360	0	1	0

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2025 · 1
Find unit circle exact value
2024 · 2
Solve for right triangle side length

What's Next

Right triangle trigonometry and the unit circle are the foundation for all further trigonometry in IB AA SL. Understanding the definitions of sine, cosine and tangent from the unit circle will let you work with trigonometric identities, graphs, and equations, all of which are heavily tested in exams. Exact values from special triangles and the unit circle are almost always required in Paper 1 non-calculator questions, so mastering this topic early will save you time and points later. Next, you will build on these core definitions to explore more advanced trigonometric topics.