Aim: Co-functions & Quotient Identities Course: Alg. 2 & Trig.

Aim: What are the Co-functions and Quotient Identities in Trigonometry?

Do Now:

cos adjacenthypotenuse

? tan= opposite

adjacent

Aim: Co-functions & Quotient Identities Course: Alg. 2 & Trig.

Reciprocal Identities

csc 1

sin

sec 1

cos

cot 1

tan

Quotient Identities

1sin

csc

1cos

sec

1tan

cot

sintan

cos

cos

cotsin

Use in verification of identities and in numerical calculations

Aim: Co-functions & Quotient Identities Course: Alg. 2 & Trig.

Verify the Identity

sin cot cos

1

tasin c

nos

A trig identity is an equation that is true for all values of variable.

verify

csc 1

sin

sec 1

cos

cot 1

tan

sintan

cos

coscot

sin

1si

sin con

cos

s

cos

ssin cos

in

cos cos

Aim: Co-functions & Quotient Identities Course: Alg. 2 & Trig.

What does simplifying mean?

Simplify the expression csc tan .

csc tan 1 sin

sin cos

1

cos sec

csc 1

sinReciprocal Identity

sintan

cos

Quotient Identity

Simplifying a trig expressions usually means expressing in terms of sine and cosine.

Aim: Co-functions & Quotient Identities Course: Alg. 2 & Trig.

Model Problems

csc 1

sin

sec 1

cos

cot 1

tanWrite an expression in terms of sin , cos , or both. Simplify whenever possible.

1. sec • cot

1

cos

1

tan

1

cos

cossin

1

sin

2.

sincos

1sin

tancsc

cos sincos sin

sin2cos

(sin )2 = sin2 usually written

Aim: Co-functions & Quotient Identities Course: Alg. 2 & Trig.

Model Problems

csc 1

sin

sec 1

cos

cot 1

tan

Find the exact numerical value for each:

1. cot2 30º

1

tan2 300

1

33

2

3

1 tan ENTER

Display: 3

÷ 30

Calculator:

) x2

22cot cotx x

Aim: Co-functions & Quotient Identities Course: Alg. 2 & Trig.

Model Problems

csc 1

sin

sec 1

cos

cot 1

tan

Find the exact numerical value for each:2. cot 45º • csc 45º

tan 45º = 1

1 • cot 45º • csc 45º

0 2sin45

2

0 1csc45 2

22

cot 45º= 1

2 2

Aim: Co-functions & Quotient Identities Course: Alg. 2 & Trig.

Trigonometry Co-functions

sin = cos(90º – ) cos = sin(90º – )Co-

sine cosine

tan = cot(90º – ) cot = tan(90º – )Co-

tangent cotangent

sec = csc(90º – ) csc = sec(90º – )Co-

secant cosecant

An trigonometric function of an acuteangle is equal to the co-function of its

complement.

Aim: Co-functions & Quotient Identities Course: Alg. 2 & Trig.

Model Problems

Write in terms of the co-function the sine of 30o.

sin = cos(90º – ) cos = sin(90º – )Co-

1sin30

2 cos(90 30 ) cos60

If x and y are the measures of two acute angles and tan x = cot y, then

1) x = y + 90 2) x = y – 90

3) x = 90 – y 4) y = x – 90

tan x = cot(90º – x) cot x = tan(90º – x)Co-

y = (90º – x)

Aim: Co-functions & Quotient Identities Course: Alg. 2 & Trig.

Model Problem

If sin x = cos(x + 20º) and x and (x + 20º)are the measure of two acute angles, find x.

sine and cosine are co-functions

If the sine of an acute angleis equal to the cosine of another

acute angle, the angles mustbe complementary.

x + 20º + x = 90º2x + 20º = 90º

2x = 70ºx = 35º x + 20º = 55º

sin 35º = .57357… cos 55º = .57357… √

Aim: Co-functions & Quotient Identities Course: Alg. 2 & Trig.

Model Problems

Which expression is NOT equal to the other three expressions?

2 cot sin 2cosA. B. C. D.

1 1tan sincos2 2