Trigonometric Functions of Any Angle

Study Guide: Trigonometric Functions of Any Angle

1. Basic Definitions

• Trigonometric functions relate angles to lengths of sides in a right triangle and are defined for all angles using the unit circle.

2. Trigonometric Functions

• Sine (sin): For an angle ( \theta ), [ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} ]

• Cosine (cos): For an angle ( \theta ), [ \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} ]

• Tangent (tan): For an angle ( \theta ), [ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{\sin(\theta)}{\cos(\theta)} ]

3. Functions for All Angles

Unit Circle Approach

• The unit circle is a circle with a radius of 1, centered at the origin of a coordinate plane.

Angle Measurement

• Angles can be measured in degrees or radians.

  • Conversion: [ \text{Radians} = \frac{\text{Degrees} \times \pi}{180} ]

Function Values for Special Angles

• Common angles (0, 30, 45, 60, 90 degrees) have known sine, cosine, and tangent values.

Reference Angles

• The reference angle is the acute angle formed by the terminal side of an angle in standard position and the x-axis.

  • Use the reference angle to determine the sign of the trigonometric functions in different quadrants.

Quadrant Analysis

• Quadrants dictate the signs of sine, cosine, and tangent:

  • Quadrant I: All functions are positive.

  • Quadrant II: ( \sin ) is positive, ( \cos ) and ( \tan ) are negative.

  • Quadrant III: ( \tan ) is positive, ( \sin ) and ( \cos ) are negative.

  • Quadrant IV: ( \cos ) is positive, ( \sin ) and ( \tan ) are negative.

4. Key Trigonometric Identities

• Pythagorean Identity: [ \sin^2(\theta) + \cos^2(\theta) = 1 ]

• Reciprocal Identities:

  • [ \csc(\theta) = \frac{1}{\sin(\theta)} ]

  • [ \sec(\theta) = \frac{1}{\cos(\theta)} ]

  • [ \cot(\theta) = \frac{1}{\tan(\theta)} ]

• Angle Sum and Difference Identities:

  • [ \sin(a \pm b) = \sin(a)\cos(b) \pm \cos(a)\sin(b) ]

  • [ \cos(a \pm b) = \cos(a)\cos(b) \mp \sin(a)\sin(b) ]

  • [ \tan(a \pm b) = \frac{\tan(a) \pm \tan(b)}{1 \mp \tan(a)\tan(b)} ]

5. Applications

• Trigonometric functions are used in various fields, including physics, engineering, and computer graphics, for modeling periodic phenomena, solving triangles, and studying waves.