On this page, we practice remembering the common trigonometric values for special angles:
\(0^\circ, 30^\circ, 45^\circ, 60^\circ,\) and \(90^\circ\)
We also practice finding the values of \(\sin, \cos\), and \(\tan\) using a simple trick.
Remember the sine values:
\(\sin(0^\circ) = \sqrt\frac{0}{4} = 0\)
\(\sin(30^\circ) = \sqrt\frac{1}{4} = \frac{1}{2}\)
\(\sin(45^\circ) = \sqrt\frac{2}{4} = \frac{1}{\sqrt 2}\)
\(\sin(60^\circ) = \sqrt\frac{3}{4} = \frac{\sqrt 3}{2}\)
\(\sin(90^\circ) = \sqrt\frac{4}{4} = 1\)
Reverse sine values for cosine values:
\(\cos(0^\circ) = 1\)
\(\cos(30^\circ) = \frac{\sqrt 3}{2}\)
\(\cos(45^\circ) = \frac{1}{\sqrt 2}\)
\(\cos(60^\circ) = \frac{1}{2}\)
\(\cos(90^\circ) = 0\)
For tangent values, use:
\(\tan(\theta)=\frac{\sin(\theta)}{\cos(\theta)}\)
Example 1
Find \(\sin(45^\circ)\)
Solution:
Step 1: Use the pattern
\(\sin(45^\circ) = \sqrt{\frac{2}{4}}\)
Step 2: Simplify
\( \sqrt{\frac{2}{4}} = \frac{√2}{2}\)
So,
\(\sin(45^\circ) = \frac{√2}{2}\)
Example 2
Find \(\cos(30^\circ)\)
Solution:
Step 1: Reverse sine values
\(\cos(30^\circ) = \frac{\sqrt 3}{2}\)
Example 3
Find \(\tan(30^\circ)\)
Solution:
Step 1:
\(\tan(θ) = \frac{sin(θ)}{cos(θ)}\)
Step 2:
\(\tan(30^\circ) = \frac{1}{2} ÷ \frac{\sqrt 3}{2}\)
Step 3:
\(= \frac{1}{2} × \frac{2}{\sqrt 3}\)
Step 4:
\(= \frac{1 }{\sqrt 3}\)
Example 4
Find \(\tan(45^\circ)\)
Solution:
Step 1:
\(\tan(45^\circ) = \frac{1}{\sqrt 2} ÷ \frac{1}{\sqrt 2}\)
Step 2:
\(= 1\)
Example 5
Find \(\tan(90^\circ)\)
Solution:
Step 1:
\(\tan(90^\circ) = \frac{\sin(90^\circ)}{\cos(90^\circ)}\)
Step 2:
\(= \frac{1}{0}\)
👉 This is undefined
Practice
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