Trigonometric Functions Overview

Study Notes

Trigonometric Functions

Trigonometric functions relate angles in a right-angled triangle to ratios of its sides.
The fundamental trigonometric functions are sine (sin), cosine (cos), and tangent (tan).
Their reciprocals are cosecant (csc), secant (sec), and cotangent (cot), respectively.
The sine of an angle (sin θ) is the ratio of the side opposite to the angle to the hypotenuse.
The cosine of an angle (cos θ) is the ratio of the side adjacent to the angle to the hypotenuse.
The tangent of an angle (tan θ) is the ratio of the side opposite to the angle to the side adjacent to the angle.
These functions are defined for angles in a right-angled triangle.
Trigonometric functions can also be used to describe angles in a unit circle.
The unit circle relates angles to points (x, y) on the circle with radius 1.
In the unit circle, sin θ represents the y-coordinate and cos θ represents the x-coordinate of the point corresponding to the angle θ.

Trigonometric Identities

Trigonometric identities are equations that are true for all valid values of the variables.
Common trigonometric identities include:
- sin²θ + cos²θ = 1
- tan²θ + 1 = sec²θ
- 1 + cot²θ = csc²θ
Pythagorean identities are fundamental relationships between trigonometric functions.
These identities are used to simplify expressions and solve trigonometric equations.
Other identities involve sums and differences of angles, such as
- sin(A + B) = sinA cosB + cosA sinB
- cos(A + B) = cosA cosB – sinA sinB
These identities are used to find the trigonometric values of sums and differences of angles.

Solving Trigonometric Equations

Trigonometric equations involve trigonometric functions of a variable.
Solving these involves finding the values of the variable that make the equation true.
Techniques include using trigonometric identities, factoring, and inverse trigonometric functions.
Solutions often involve multiple angles due to the periodic nature of trigonometric functions.
Solutions must be within the specified range of angles or periodic range.

Graphs of Trigonometric Functions

The graphs of sine, cosine, tangent, and other trigonometric functions are periodic.
The graph of sin(x), for example, oscillates between -1 and 1 in a repeating pattern.
The graphs of trigonometric functions are useful for visualizing their behavior and finding solutions to trigonometric equations.
The period of sin(x) and cos(x) is 2π.
The period of tan(x) is π.
Graphs can be shifted horizontally (phase shifts) and vertically.

Inverse Trigonometric Functions

Inverse trigonometric functions, like arcsin, arccos, and arctan, are used to find the angle given the ratio of sides.
These functions are used to 'undo' the effect of the trigonometric function.
Important to note the restricted ranges for these inverse functions. For example, arccos(x) is restricted to the range 0 to π.
The domains and ranges of inverse trigonometric functions are different from the original functions.

Description

This quiz covers the fundamentals of trigonometric functions, focusing on their relationship to right-angled triangles and the unit circle. Key functions including sine, cosine, and tangent, along with their reciprocals, will be explored. Test your understanding of these essential concepts in trigonometry.