Precalculus Study Notes: Trigonometric and Circular Functions

Tangent, Cotangent, Secant, and Cosecant Functions

Definitions and Properties

These functions are fundamental trigonometric functions derived from the sine and cosine functions. They are used to describe relationships in right triangles and on the unit circle.

• Tangent (tan): Defined as the ratio of sine to cosine. $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$

• Cotangent (cot): The reciprocal of tangent. $\cot(\theta) = \frac{1}{\tan(\theta)} = \frac{\cos(\theta)}{\sin(\theta)}$

• Secant (sec): The reciprocal of cosine. $\sec(\theta) = \frac{1}{\cos(\theta)}$

• Cosecant (csc): The reciprocal of sine. $\csc(\theta) = \frac{1}{\sin(\theta)}$

Example: For $\theta = 45^\circ$, $\tan(45^\circ) = 1$, $\cot(45^\circ) = 1$, $\sec(45^\circ) = \sqrt{2}$, $\csc(45^\circ) = \sqrt{2}$.

Radian Measure of Angles

Understanding Radians

Radians are an alternative to degrees for measuring angles, based on the radius of a circle.

• Definition: One radian is the angle subtended at the center of a circle by an arc whose length equals the radius of the circle.

• Conversion: $180^\circ = \pi$ radians

• Formula: $\text{Angle in radians} = \frac{\text{Angle in degrees} \times \pi}{180}$

Example: $90^\circ = \frac{\pi}{2}$ radians

Circular Functions

Definition and Applications

Circular functions are trigonometric functions defined using the unit circle. They describe the coordinates of points on the circle as a function of the angle.

• Sine and Cosine: For an angle $\theta$, the coordinates on the unit circle are $(\cos(\theta), \sin(\theta))$.

• Periodicity: All circular functions are periodic, repeating every $2\pi$ radians.

Example: $\sin(\theta)$ and $\cos(\theta)$ oscillate between -1 and 1 as $\theta$ increases.

Inverse Circular Relations

Inverse Trigonometric Functions

Inverse trigonometric functions allow us to find angles given the value of a trigonometric function.

• Arcsin: $\arcsin(x)$ gives the angle whose sine is $x$.

• Arccos: $\arccos(x)$ gives the angle whose cosine is $x$.

• Arctan: $\arctan(x)$ gives the angle whose tangent is $x$.

Example: $\arcsin(0.5) = 30^\circ$ or $\frac{\pi}{6}$ radians.

Sinusoidal Functions as Mathematical Models

Modeling with Sine and Cosine

Sinusoidal functions are used to model periodic phenomena such as sound waves, tides, and seasonal temperatures.

• General Form: $y = A \sin(Bx + C) + D$

• Parameters:

  • A: Amplitude (height of wave)

  • B: Frequency (number of cycles per unit)

  • C: Phase shift (horizontal shift)

  • D: Vertical shift

Example: The function $y = 3 \sin(2x - \frac{\pi}{4}) + 1$ models a wave with amplitude 3, frequency 2, phase shift $\frac{\pi}{4}$, and vertical shift 1.

Additional info: The original notes were brief and fragmented; academic context and examples have been added for completeness.