aim: what are the reciprocal functions and cofunction?

Aim: What are the reciprocal functions and cofunction?

Do Now: In ,90, CABC AB = 17 and BC = 15.

1) Find

Asin Acos Atana) AC b)

c) d)

2) Find the reciprocal ofAsin Acos Atana) b) c)

A

BC

Worksheet

Asin Acos

Atan

1) a) AC = 8, b) = 15/17 c) = 8/17

= 15/8d)

Asin Acos

Atan

2) a) rec. of = 17/15 b) rec. of

c) rec. of = 8/15

= 17/8

When we write the reciprocal of sine, cosine and tangent, we get 3 other functions – the reciprocal functions.

We have names for those three reciprocal functions

adjacent

hypotenusesec

Definition of the reciprocal functions

opposite

hypotenusecsc

opposite

adjacentcot

sin

1csc

cos

1sec

tan

1cot

or

csc

1sin

or

or

sec

1cos

cot

1tan

Quadrant I II III IV sin + + – –

csc + + – –cos + – – +sec + – – +tan + – + –cot + – + –

Properties of all 6 trig functions

A

B C

30

Given ΔABC, mB = 30, find the values of a) sin B b) cos A

12

32

130sin

2

160cos

Then sin30 = cos60

That is sin30 = cos (90 – 30)

In general, sin A = cos (90 – A) or cos A = sin (90 – A)

The sine of an acute angle is equal to the cosine of its complement. The sine and cosine are cofunctions

tan A = cot (90 – A) cot A = tan (90 – A)

sec A = csc (90 – A) csc A = sec (90 – A)

tangent and cotangent are cofunctions

secant and cosecant are cofunctions

In all three pairs of cofunctions, the angles are complementary angles.

Relationship of all trig functions

sin

cos

tan

cot

sec

csc

cofunctions

cofunctions & reciprocal

cofunctions

reciprocal

reciprocal

Find a value of for which the statement is true.

1. cos17sin

2. )60cot(tan

3. )83csc()10sec(

4. )242cos(sin

sin cosWrite the following functions in terms of

, or both:

cot

sec

tan

csc

sin

a)

b)

c)

d)

150csc

240sec

3cot

3sec

3

2cot

Find the values of the followinga)

b)

c)

d)

e)

Determine the quadrant of the angle lie

0csc

0cot

0cot 0sin

0sec 0tan

a)

b)

c) and

d) and