1. the Reciprocal Trigonometric Functions

TRIGONOMETRY

1. The Reciprocal Trigonometric Functions

We define the secant, cosecant and cotangent functions as

Activity1:Beginning with the graphs of cosine, sine and tangent, obtain the graphs of secant, cosecant and cotangent then check on Autograph or a graphic calculator.

Example1: Find, in surd form,
a)

b)

c)

a) / / b) / / c) /

C3 p75 Ex 6A, p78 Ex 6B

Example 2 : Simplify the following.

a)

b)

c)

a) / / b) /
c) /

Example 3 : Solve the following equations in the interval .

a)

b)

c)

d)

a) / /
b) /
c) / /
d) / /

C3 p81 Ex 6C Q1-2, 5-8

2. Pythagorean Relationships

First a review of the Pythagorean relationship from unit C2. Consider the right-angled triangle with hypotenuse 1. The opposite has length and the adjacent has length .

Using Pythagoras’ rule we have

We can obtain two further Pythagorean relationships from this identity.

Dividing through by ,

And dividing through by ,

With these three Pythagorean relationships, we can prove many other identities and solve certain trigonometric equations.

Example 1 : Prove the following identities.

a)

b)

c)

d)

It is usual to take the more ‘complicated’ side (often the LHS), and reduce it to the form of the ‘simpler’ side.

a) /
b) /
c) /
d) /

Example 2 : Solve the following equations for all values of θ between 0° and 360°.

a)

b)

c)

d)

These problems are solved by using identities to eliminate all but one of the trigonometric functions.

a) / /
b) / /
c) / /
d) /

Example 3 : Given that and that θ is obtuse, find the exact value of

a)

b)

a) /
We choose the negative root since cos is negative for obtuse angles, and therefore so is sec. / b) /

C3p81 Ex 6C Q3-4 p85 Ex 6D

3. Inverse Trigonometric Functions

Consider the inverse function of . This is known as or . Its graph can be obtained by reflecting the graph of in the line .

However, since is a many-to-one function, its inverse will be one-to-many, and therefore will not be a function. We get round this problem by restricting the domain of to, so it becomes a one-to-one function. Its inverse is then also one-to-one.

If you use the inverse sine button on a calculator, it will always give an angle in the range, or if you are working in degrees.

In a similar way, we make a function by restricting the domain of to, and a function by restricting the domain of to.

Note the horizontal asymptotes on the graph.

Example 1 : Find, in radians, the values of

a)

b)

c)

a) / /
b) / /
c) / /

C3 p90 Ex 6E

4. Compound Angle Formulas

We will derive expressions for and.
Consider the diagram opposite.

In a similar fashion,

Replacing B by −B in these identities,

We also have

We arrived at the last line by dividing top and bottom by .

Replacing B by −Bin this identity,

We can summarise the compound angle formulas as follows.

Example 1 : Evaluate the following in surd form, without the use of a calculator.

a) cos 75°

b) tan 15°

a) /
b) /

Example2:Given that is acute, and , and that is acute with , find

a)

b)

Since and are acute, we can draw right-angled triangles including them.

a) /
b) /

We can verify these answers by actually finding and .

Example3:Given that is acute, and, and that is obtuse with , find

Imagine first that both and are acute.

Since is obtuse, we have , instead of . Note that we are effectively solving , for which there is a positive and a negative solution, dependent on which quadrant lies in.

Example 4 : Prove the following identities.

a)

b)

c)

d)

a) /
b) /
c) /
d) /
To get the last line, we divide top and bottom by .

Example 5 : Solve the following equations for .

a)

b)

c)

a) / /
b) /
c) /

C3 p99 Ex 7A

5. Double Angle Formulas

We already have the compound angle formulas.

Putting B = A we obtain the double angle formulas. Notice that we can use the Pythagorean relationship to obtain three identities for .

These are so called because the angle on the LHS of the identity is double the angle on the RHS. So we can also say, for example,

Example1:If is acute and such that , find

a)

b)

c)

a) / / b) / / c) /

Example 2 : Prove that

a) c)

b) d)

a) / / c) /
b) / / d) /

Example 3 : Solve the following equations for .

a) .

b) .

c) .

a) / /
b) /
c) / /

C3 p103 Ex 7B, p106 Ex 7C

6. Identities for

Example 1 : Solve the equation for .

It is not immediately obvious how to solve this equation. However, consider the compound angle identity...

If we now write...

...and compare coefficients of and, we have the equations...

Dividing,

Squaring the two equations and then adding,

So the equation becomes

At this stage, notice that we have found out that

This means that if we translate the graph of by 56.3° to the left, and then stretch vertically with a scale factor of , we get the graph of So this method also allows us to sketch trigonometric functions of the form .

There are four compound angle formulas, to be used in certain conditions...

/ used for /
/ used for /
/ used for /
/ used for /

In other words, the first and last are interchangeable.

Activity 2 : So now try example 1 with !

Example2:Find the greatest and least values of , and the smallest positive values of x for which these occur.

Comparing coefficients,

Dividing,

Squaring the two equations and then adding,

So we want to find the greatest and least values of .

The greatest value is 17, and this occurs when

The least value is –17, and this occurs when

Example 3 : Express in the form , where is in radians.

a) Find the greatest and least values of

and the least positive values of x for which they occur.

b)Determine, in radians to 2 decimal places, the values of x in the interval [0, 2π] for which

Using the identity

Comparing coefficients,

This gives us

a)This expression takes all values between –5 and 5. Therefore the minimum value of the expression is given by

This happens when

The maximum value is

This happens when

b) We have

Example4:The diagram shows the plan view of a table, measuring 80cm by 120 cm, stuck in a corridor 130 cm wide. Find the angle θ. There are two answers.

Considering the dotted line,

Using the identity

Comparing coefficients,

This gives us

And so our equation becomes

C3 p111 Ex 7D

7. The Factor Formulas

Consider the two compound angle formulas

Adding these identities,

Now let and . Substituting,

Notice we have now expressed the sum of two trigonometric quantities as the product of two others. This is known as a factor formula for this reason.

If instead we subtract the compound angle formulas instead of adding, we have

Now consider the cosine compound angle formulas.

Adding these identities,

If instead we subtract the compound angle formulas instead of adding, we have

So our four factor formulas are

Example1:Prove the identity

Using the second and third factor formulas,

Example 2 : Find, in surd form, the value of .

Example 3 : Solve the equation for .

So the solutions are

Example 4 : Solve the equation for .

So the solutions are.

C3 p115 Ex 7E Topic Review : Trigonometry