c3 Trigonometry

C3 Trigonometry - 1 -

Secant, Cosecant, Cotangent

1. Sec x = , cos x 0

2. Cosec x = , sin x 0

3. Cot x = , tan x 0

= , sin x 0

On your calculator, you will have to use sin, cos and tan keys, followed by the

reciprocal key . However, to find exact values for the 'special'

angles, follow the following examples.

Examples

Find the exact value of:

1 (i) sec 60(ii) cosec 60(iii) cot 30(iv) cosec 45

2. (i) cosec

(ii) sec

(iii) cot

Solution

With calculaor in degrees

1. (i) Press cos to get 0.5 then press 1/x to get 2

(ii) press sin 60 to get 018660 ... which isn't a recognisable fraction ...

then press to get 0.75

i.e. sin2 60 =

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x2


so, sin 60 =

so, cosec 60 =

(iii) press tan 30 to get 05773 ... which isn’t a recognisable fraction ... then press to get 0.3333 ... which is .

i.e. tan2 30 =

so, tan 30 =

so, cot 30 =

(iv) press sin 45 to get 0.7071 ... which is not a recognisable fraction ...

then press to get 0.5

i.e. sin2 45 =

so, sin 45 =

so, cosec 45 =

with calculator in rads:

2. (i) press sin ( 6) to get 0.5 cosec = 2

(ii) press cos ( 4) to get 0.7071 ... then press to get 0.5 .

i.e. cos2 =

so, cos =

sec, =


x2

x2

x2


(iii) Press tan to get 1.732 ... then press to get 3

i.e. tan2 = 3

tan =

cot =

The graphs of sec, cosec and cot are shown below:


x2


Their period are the same as cos, sin and tan.

Also,

Trigonometric Identities

You already know sin2 + Cos2 1

If you divide by Cos2 +1

We get

similarly, dividing by sin2 gives

These can be used to simplify expressions


Sin+ All+

Tan+ Cos+

Cosec+ All+

Cot+ Sec+

tan2 + 1 sec2

1 + cot2


Example

(i) Express 5cot + 2 cosec2 in terms of cot

(ii) Solve the equation 5 cot + 2 cosec2 = 5 for 0 < < 2

Solution

(i) 5 cot + 2 cosec2 5 cot + 2 (1 + cot2 ) 5 cot + 2 + 2cot2

(ii) 5 cot + 2 cosec2 = 5

5 cot + 2 + 2 cot2 = 5 2 cot2 + 5 cot - 3 = 0

This is a quadratic in cot

cot =

=

tan = 2 or -

Because the range of is given in terms of we must have ourCalculator in radians


+ve tan 1st and 3rd quad.

-ve tan 2nd and 4th quad


The Addition Formulae

For all angles, A and B,

(these are in the formula book)

Example 1: Prove the identity

Solution:



Example 2: By writing 75o as (30o + 45o) find the exact value of cos75o

Solution: cos75 = cos(30 + 45) = cos30cos45 - sin30sin45

Example 3: Find the value of tanxo given that sin(x + 30o) = 2cos(x – 30o)

Solution: sin(x + 30) = 2cos(x - 30)

Rearranging gives

Double Angle Formulae(LEARN)

Putting A=B into the addition formulae gives identities for sin2A, cos2A and tan2A:-

Using, sin(A+B) = sin A cos B + cos A sin B.

Sin 2A = sin(A+A) = sin A cos A + cos A sin

Sin 2A = 2 sin A cos A



Similarly

Note: there are 3 forms of cos2A

and

tan 2A =

Example: Solve the equation

Solution:

4 sin sin

or



Example; If sin A = and A is obtuse, find the exact values of cos A,

sin2 A, tan 2 A.Solution:

Example: Solve cos 2A + 3 + 4cos A = 0,

Solution: (Since we have cos A in the equation we choose the form of cos 2A which involves cos A only)

i

(this is a quadratic in cosA)



It is useful to be able to express the sum of sin x and cos x as a single sin or cos. It enables us to:

- find the max/min of the expression- solve equations

To find R and , we expand the right hand side and equate like terms.

This gives R =

and values for sin and cos in terms of a, b and R.

Example: i) Express

ii) Find the max and min of the expression and find in radians to 2dp the smallest positive value of for which they occur.

iii) Solve the equation Solution:

(i) R=

Equating Equating



NB: 3sin x + 2cos x can also be expressed in the form R cos ( ), but it is usually more convenient to choose the form which produces the terms in the right order with the correct sign.

i.e

Inverse Trig Functions

For to exist then must be a one to one function.

If Is not 1-to-1 over its whole domain then can exist

provided we restrict the domain of

The functions and are not 1-to-1 over their whole domain.

The inverse functions and exist for the domains:



The graph of (x) is a reflection of f(x) in the line y = x

are reflections of sin x, cos x tan x in the line y = x (x in radians, same scale on both axes).


c3 Trigonometry

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