To evaluate trigonometric functions in exact form means finding their exact values rather than decimal approximations. This involves using known values, angles, and trigonometric identities.

Here’s a step-by-step guide to help you evaluate trig functions in exact form:

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1. Know Special Angles and Their Values

Many trig functions have exact values for specific angles commonly used in trigonometry, such as:

Angle (Degrees)	Angle (Radians)	sin θ	cos θ	tan θ
0°	0	0	1	0
30°	π/6	1/2	√3/2	1/√3
45°	π/4	√2/2	√2/2	1
60°	π/3	√3/2	1/2	√3
90°	π/2	1	0	Undefined

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2. Use the Unit Circle

The unit circle provides exact values for sine, cosine, and tangent for common angles in all four quadrants.

For example:

• $\sin(\pi/3)$ corresponds to the y-coordinate at $(\pi/3)$ on the unit circle, which is $\sqrt{3}/2$.
• $\cos(\pi/4)$ corresponds to the x-coordinate at $(\pi/4)$, which is $\sqrt{2}/2$.

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3. Trigonometric Identities

Some key identities can help simplify expressions and find exact values:

• Reciprocal Identities:

  • $\csc \theta = \frac{1}{\sin \theta}$
  • $\sec \theta = \frac{1}{\cos \theta}$
  • $\cot \theta = \frac{1}{\tan \theta}$

• Pythagorean Identities:

  • $\sin^2 \theta + \cos^2 \theta = 1$
  • $1 + \tan^2 \theta = \sec^2 \theta$
  • $1 + \cot^2 \theta = \csc^2 \theta$

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4. Reference Angles

If the angle is not in the first quadrant, use reference angles to determine the exact value.

• Reference angle: The acute angle ($\theta$) formed between the terminal side and the $x$-axis.
• For angles in:
  • Quadrant II: $\theta' = 180^\circ - \theta$ or $\pi - \theta$
  • Quadrant III: $\theta' = \theta - 180^\circ$ or $\theta - \pi$
  • Quadrant IV: $\theta' = 360^\circ - \theta$ or $2\pi - \theta$

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5. Exact Values for Special Triangles

• 30°-60°-90° Triangle:

  • Opposite 30°: $1$
  • Opposite 60°: $\sqrt{3}$
  • Hypotenuse: $2$

• 45°-45°-90° Triangle:

  • Opposite each 45° angle: $1$
  • Hypotenuse: $\sqrt{2}$

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Example Calculations

1. $\sin(60^\circ)$

   • From the table or unit circle:
     $\sin(60^\circ) = \sqrt{3}/2$

2. $\cos(225^\circ)$

   • $225^\circ$ is in Quadrant III.
   • Reference angle: $225^\circ - 180^\circ = 45^\circ$.
     $\cos(225^\circ) = -\cos(45^\circ) = -\sqrt{2}/2$

3. $\tan(\pi/4)$

   • From the table:
     $\tan(\pi/4) = 1$

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Would you like more details or additional examples?

Related Questions

1. How do you use the unit circle to find exact values?
2. What are the exact values for trigonometric functions of $120^\circ$?
3. How do you find exact values of trigonometric functions using identities?
4. How do reference angles help evaluate trig functions?
5. What is the significance of the 30°-60°-90° and 45°-45°-90° triangles?

Tip: Memorize the unit circle and common angle values to quickly evaluate trig functions in exact form.