Trigonometric functions relate angles to lengths of sides in a right triangle and are defined for all angles using the unit circle.
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)} ]
The unit circle is a circle with a radius of 1, centered at the origin of a coordinate plane.
Angles can be measured in degrees or radians.
Conversion: [ \text{Radians} = \frac{\text{Degrees} \times \pi}{180} ]
Common angles (0, 30, 45, 60, 90 degrees) have known sine, cosine, and tangent values.
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.
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.
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)} ]
Trigonometric functions are used in various fields, including physics, engineering, and computer graphics, for modeling periodic phenomena, solving triangles, and studying waves.