5.1 Angles and Their Measure

Degree Measure

In geometry, an angle is defined as the set of points determined by two rays with a common endpoint. The measure of an angle describes the amount of rotation from one ray (the initial side) to the other (the terminal side).

• Initial side: The starting ray of the angle.

• Terminal side: The ending ray after rotation.

• Coterminal angles: Angles that share the same initial and terminal sides but may differ by full rotations.

• Straight angle: An angle of .

• Standard position: An angle whose initial side lies along the positive x-axis.

• Positive angle: Rotation is counterclockwise.

• Negative angle: Rotation is clockwise.

Example: Find two coterminal angles, one positive and one negative, to the angle .

• Add or subtract to find coterminal angles: (positive), (negative).

One unit of measurement for angles is the degree. The angle in standard position obtained by one complete revolution in the counterclockwise direction has measure .

Angle Terminology

• Right angle:

• Acute angle:

• Obtuse angle:

• Complementary angles: Two angles whose measures add up to

• Supplementary angles: Two angles whose measures add up to

Example: Find the supplementary angles for :

• a)

• b)

• c)

Definition of Radian Measure

One radian is the measure of the central angle of a circle subtended by an arc equal in length to the radius of the circle.

• a) radian

• b) radians

• c) radians

• d) radians

Relationship Between Degrees and Radians

• radians

• radians radians

• $1= \frac{180^{\circ}}{\pi} \approx 57.2958^{\circ}$

When radian measure of an angle is used, no units will be indicated.

Changing Angular Measures

• To change degrees to radians, multiply by

• To change radians to degrees, multiply by

Students should learn the radian measure for the standard angles , and multiples up to .

Example: Change to radians:

Formula for the Length of a Circular Arc

If an arc of length on a circle of radius subtends a central angle of radian measure , then:

5.2 Right Triangle Trigonometry

Definition and Properties

Trigonometric functions are introduced as ratios of sides of a right triangle. A right triangle is a triangle with a right angle ().

• Hypotenuse (hyp): The side opposite the right angle, always the longest side.

• Opposite (opp): The side opposite the angle .

• Adjacent (adj): The side next to the angle (not the hypotenuse).

Definition of the Trigonometric Functions of an Acute Angle of a Right Triangle

The domain of each trig function is all real numbers for which the denominator is not zero. The values of the six trigonometric functions are positive for every acute angle .

Reciprocal Identities

Fundamental Identities

• Pythagorean identities:

• Tangent and cotangent identities:

Special Values of the Trigonometric Functions

The following table lists the values of the six trigonometric functions for commonly used angles:

Additional info: Table entries for undefined values are due to division by zero in the trigonometric ratios.

Examples and Applications

• Example: In a right triangle, , opposite side = 3, adjacent side = x, hypotenuse = y. Find , , , , , .

• Example: Distance to Mt. Fuji. The peak of Mt. Fuji in Japan is approximately 12,400 ft high. Estimate the distance from the student to the point on level ground directly beneath the peak, given the angle between level ground and the peak is .

Co-function Identities

Co-function identities relate the trigonometric functions of complementary angles ( and ).

Example: Given , find the values of the other trigonometric functions of .