Special Angle Values
30°, 45° and 60° have exact trig values.
About this topic
Exact values of sine, cosine and tangent at 0°, 30°, 45°, 60° and 90° come from the special right triangles. For example sin 30° = 1/2, sin 45° = √2/2 and sin 60° = √3/2, with cosines running in reverse order.
The 45-45-90 triangle has sides 1, 1 and √2, and the 30-60-90 triangle has sides 1, √3 and 2, obtained by cutting an equilateral triangle of side 2 in half. Reading ratios off these triangles gives the exact values with no calculator. Tangent values are 0, √3/3, 1, √3 and undefined at 90°.
Using symmetry on the unit circle extends these values to every multiple of 30° and 45°. For instance cos 120° = −1/2 because 120° has a reference angle of 60° and cosine is negative in the second quadrant. A handy memory trick is that sine of 0°, 30°, 45°, 60°, 90° equals the square root of 0, 1, 2, 3, 4 divided by 2.
Exact values matter in physics, engineering and geometry proofs, where decimals would hide patterns. A common error is mixing up sin 30° and sin 60°; remember that a small angle has a small sine, so sin 30° must be the smaller value of 1/2.
Key ideas
- sin 30° = 1/2
- sin 45° = √2/2
- sin 60° = √3/2
Quick check
Use cases: where Special Angle Values helps
- Homework: review the key idea behind Special Angle Values before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Special Angle Values without notes.
- Visual learning: play with the interactive lab to see how Special Angle Values changes when you adjust its inputs.
- Teaching a friend: explain Special Angle Values in your own words, then compare your version with the key ideas.
Number and algebra skills are used in science, engineering, budgeting and computing.
Frequently asked questions
What is Special Angle Values?
30°, 45° and 60° have exact trig values.
What will I learn in Special Angle Values?
sin 30° = 1/2. sin 45° = √2/2. sin 60° = √3/2.
Is Special Angle Values part of Mathematics?
Yes. It is one of 200 Mathematics topics on LearnViz HighSchoolEdu, with 3 key ideas, a live animation and a quick check.
Does the Special Angle Values lab work on a phone or tablet?
Yes. The animation resizes to fit your screen and supports touch, so you can drag, tap and slide with your finger.
How should I study Special Angle Values?
Read the summary and key ideas first, play with the interactive lab to see the idea in motion, then answer the quick check. If you get a question wrong, go back to the key idea it tests.
Do I need an account, and is my progress saved?
No account is needed. Marking a topic as learned is saved in your own browser, so it stays on that device and is not sent to a server.