y x Radian: The length of the arc above the angle divided by the radius of the circle. Definition, in radians

y x Definition, in radians Unit Circle: the circle with radius of 1 unit If r =1, = s 1

Definition The radian measure of an angle is the distance traveled around the unit circle. Since circumference of a circle is 2 r and r =1, the distance around the unit circle is 2

Important Idea If a circle contains 360° or 2  radians, how many radians are in 180° Use to change rads to degrees 180° rads Use to change degrees to rads rads 180°

Try This Change 240° to radian measure in terms of .

Try This Change radians to degree measure °

Try This Change radians to degree measure °

Definition Initial Side Terminal Side Vertex Angle A is in standard position A x y

Definition A If the terminal side moves counter- clockwise, angle A is positive x y

Definition A If the terminal side moves counter- clockwise, angle A is positive x y

Definition A If the terminal side moves counter- clockwise, angle A is positive x y

Definition A If the terminal side moves clockwise, angle A is negative x y

Definition A If the terminal side moves clockwise, angle A is negative x y

Definition A If the terminal side moves clockwise, angle A is negative x y

Definition A If the terminal side moves clockwise, angle A is negative x y

Definition A If the terminal side is on an axis, angle A is a quadrantel angle x y

Definition A If the terminal side is on an axis, angle A is a quadrantel angle x y

Definition A If the terminal side is on an axis, angle A is a quadrantel angle x y

Definition A If the terminal side is on an axis, angle A is a quadrantel angle x y

Definition The quadrantal angles in radians

Definition The quadrantal angles in radians

Definition The quadrantal angles in radians

Definition The quadrantal angles in radians The terminal side is on an axis.

Definition Coterminal Angles: Angles that have the same terminal side. Important Idea In precal, angles can be larger than 360° or 2 radians.

Important Idea To find coterminal angles, simply add or subtract either 360° or 2 radians to the given angle or any angle that is already coterminal to the given angle.

Analysis 30° and 390° have the same terminal side, therefore, the angles are coterminal 30° x y x y 390°

Analysis 30° and 750° have the same terminal side, therefore, the angles are coterminal 30° x y x y 750°

Analysis 30° and 1110° have the same terminal side, therefore, the angles are coterminal 30° x y x y 1110°

Analysis 30° and -330° have the same terminal side, therefore, the angles are coterminal 30° x y x y -330°

Try This Find 3 angles coterminal with 60° 420°,780° and -300°

Try This Find two positive angle and one negative angle coterminal with radians. and,

Important Idea opp hyp adj opp adj r > 0

Find sin, cos & tan of the angle whose terminal side passes through the point (5,-12) Try This (5,-12)

Solution (5,-12)

Important Idea Trig ratios may be positive or negative

Find the exact value of the sin, cos and tan of the given angle in standard position. Do not use a calculator.

Solution 2

Definition Reference Angle: the acute angle between the terminal side of an angle and the x axis. (Note: x axis; not y axis). Reference angles are always positive.

Important Idea How you find the reference angle depends on which quadrant contains the given angle. The ability to quickly and accurately find a reference angle is going to be important in future lessons.

Example Find the reference angle if the given angle is 20°. In quad. 1, the given angle & the ref. angle are the same. x y 20°

Example Find the reference angle if the given angle is 120°. For given angles in quad. 2, the ref. angle is 180° less the given angle. ? 120° x y

Example Find the reference angle if the given angle is. x y For given angles in quad. 3, the ref. angle is the given angle less

Try This Find the reference angle if the given angle is For given angles in quad. 4, the ref. angle is less the given angle.

Important Idea The trig ratio of a given angle is the same as the trig ratio of its reference angle except, possibly, for the sign. Example:

The unit circle is a circle with radius of 1. We use the unit circle to find trig functions of quadrantal angles. 1 Definition

The unit circle 1 Definition (1,0) (0,1) (-1,0) (0,-1) x y

Definition (1,0) (0,1) (-1,0) (0,-1) For the quadrantal angles: The x values are the terminal sides for the cos function.

Definition (1,0) (0,1) (-1,0) (0,-1) For the quadrantal angles: The y values are the terminal sides for the sin function.

Definition (1,0) (0,1) (-1,0) (0,-1) For the quadrantal angles : The tan function is the y divided by the x

Find the values of the 6 trig functions of the quadrantal angle in standard position: Example 0° (1,0) (0,1) (-1,0) (0,-1)

Find the values of the 6 trig functions of the quadrantal angle in standard position: Example  90° (1,0) (0,1) (-1,0) (0,-1)

Find the values of the 6 trig functions of the quadrantal angle in standard position: Example 180° (1,0) (0,1) (-1,0) (0,-1)

Find the values of the 6 trig functions of the quadrantal angle in standard position: Example 270° (1,0) (0,1) (-1,0) (0,-1)

Find the values of the 6 trig functions of the quadrantal angle in standard position: Try This 360° (1,0) (0,1) (-1,0) (0,-1)

A trigonometric identity is a statement of equality between two expressions. It means one expression can be used in place of the other. A list of the basic identities can be found on p.317 of your text.

Reciprocal Identities:

Quotient Identities:

 x y r but… therefore

Divide by to get: Pythagorean Identities:

Divide by to get:

Try This Use the Identities to simplify the given expression: 1

Try This Use the Identities to simplify the given expression:

Prove that this is an identity

Now prove that this is an identity

One More