Learning the table values of $\sin$, $\cos$, and $\tan$ for common angles (0°, 30°, 45°, 60°, and 90°) is straightforward if you follow these simple steps:

1. Understand the key angles and the table structure:

You will memorize the values for these angles: $0^\circ, 30^\circ, 45^\circ, 60^\circ, 90^\circ$.

2. Sine values:

To remember $\sin$ values:

• Start with 0.
• Use the sequence: $\sin 0^\circ = 0, \sin 30^\circ = \frac{1}{2}, \sin 45^\circ = \frac{\sqrt{2}}{2}, \sin 60^\circ = \frac{\sqrt{3}}{2}, \sin 90^\circ = 1$.
• Note the pattern: $\sin \theta$ increases gradually from 0 to 1.

3. Cosine values:

To derive $\cos$ values:

• Use the reverse of $\sin$ values: $\cos \theta = \sin (90^\circ - \theta)$.
• Values are: $\cos 0^\circ = 1, \cos 30^\circ = \frac{\sqrt{3}}{2}, \cos 45^\circ = \frac{\sqrt{2}}{2}, \cos 60^\circ = \frac{1}{2}, \cos 90^\circ = 0$.

4. Tangent values:

$\tan \theta = \frac{\sin \theta}{\cos \theta}$.

• Values:
  • $\tan 0^\circ = 0$,
  • $\tan 30^\circ = \frac{1}{\sqrt{3}}$,
  • $\tan 45^\circ = 1$,
  • $\tan 60^\circ = \sqrt{3}$,
  • $\tan 90^\circ = \text{undefined}$.

5. Memorize using a table:

Here’s the quick-reference table for these values:

$\theta$	$\sin \theta$	$\cos \theta$	$\tan \theta$
$0^\circ$	$0$	$1$	$0$
$30^\circ$	$\frac{1}{2}$	$\frac{\sqrt{3}}{2}$	$\frac{1}{\sqrt{3}}$
$45^\circ$	$\frac{\sqrt{2}}{2}$	$\frac{\sqrt{2}}{2}$	$1$
$60^\circ$	$\frac{\sqrt{3}}{2}$	$\frac{1}{2}$	$\sqrt{3}$
$90^\circ$	$1$	$0$	$\text{undefined}$

6. Tips to practice:

• Write this table repeatedly.
• Solve problems using these values.
• Use mnemonic techniques to remember patterns.

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Would you like examples or additional exercises to reinforce this concept?

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Five questions to explore further:

1. Why is $\tan 90^\circ$ undefined, and how does it relate to its graph?
2. How do you use these values to find trigonometric functions of larger angles?
3. What are the reciprocal trigonometric functions, and how do they relate to $\sin$, $\cos$, and $\tan$?
4. How can you derive these values geometrically using a unit circle?
5. Can you find the exact trigonometric values of non-standard angles like 15° or 75°?

Tip: Regularly practice with real-world problems to internalize the values better