Important Idea r > 0 opp hyp adj hyp opp adj

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

Solution 5 -12 13 (5,-12)

Important Idea Trig ratios may be positive or negative

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 important.

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

Example x 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. y 120° ? x

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

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.

Definition We can use the unit circle to find trig functions of quadrantal angles. 1

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

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

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

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

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

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

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

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

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

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:

r y  but… x therefore

Pythagorean Identities: 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