# IB Maths HL – Trigonometry

### 1. IB Mathematics HL– Trigonometry, Arc Length

How can we find the length of an arc with central angle in a circle with radius 4 cm?

Solution

Let (in radians) be the central angle in a circle with radius r, then the length of the arc cut off  by is given by the following formula:

Therefore the arc length in our case is

### 2. IB Mathematics HL– Trigonometry, Area of a sector

How can we find the area of a sector with central angle in a circle with radius 4 cm?

Solution

IB Maths HL – Trigonometry, Area of a sector

Let (in radians) be the central angle in a circle with radius r, then the area of the sector formed  by angle is given by the following formula:

Therefore the arc length in our case is

### 3. IB Maths HL Trigonometric equations

How can we solve the following trigonometric equation?

for

Solution

Your primary goal in solving a trigonometric equation is to isolate the trigonometric function in the equation.

The above equation has two solutions in the interval

### 4. IB Mathematics HL– Trigonometry, Trigonometric equations

How can we solve the following trigonometric equation?

for

Solution

IB Maths HL – Trigonometry, Trigonometric equations

By treating the equation as a quadratic in and then factoring.

Now, you have to solve two trigonometric equations

and    in the interval

For the first equation we have:

The above equation has only one solution in the interval

For the second equation we have:

The above equation has two solutions in the interval

### 5. IB Mathematics HL – Trigonometry, Sine Rule

How can we find the side of a triangle ABC given that ?

Solution

IB Maths HL – Trigonometry, Sine Rule

The Sine rule states

So in your case the side can be calculated as follows:

### 6. IB Mathematics HL – Trigonometry, Cosine Rule

How can we find the side of a triangle ABC given that ?

Solution

IB Maths HL – Trigonometry, Cosine Rule

The Cosine rule states

So in your case the side can be calculated as follows:

### 7. IB Mathematics HL – Trigonometry, Area of a triangle

How can we find the area of a triangle ABC with sides

and included angle ?

Solution

IB Maths HL – Trigonometry, Area of a triangle

The area of a triangle is a half of the product of two sides and the sine of the included angle.

Therefore, in this case, we have the following

### 8. IB Mathematics HL – Trigonometry, Trigonometric identities

How can we prove the following trigonometric equality?

Solution

IB Maths HL – Trigonometry, Trigonometric identities

L.H.S

R.H.S

### 9. IB Mathematics HL – Trigonometry, Cosine Rule

How can we solve the triangle using Cosine Rule given that , and .

Solution

IB Math HL – Trigonometry, Cosine Rule

Applying the Cosine Rule we have that

Now, in order to find another angle, for example , we could apply again cosine rule and solve for as following

Finally, for the angle we have that

### 10. IB Mathematics HL – Trigonometry, Trigonometric equations

How can we solve the following trigonometric equation?

in the interval .

Solution

IB Math HL – Trigonometry, Trigonometric equations

From unit circle we have the following:

accepted

or  rejected

or accepted

or accepted

or  accepted

or accepted

or  rejected

Therefore, the solutions are

### 11. IB Mathematics HL – Trigonometry, Trigonometric equations, tanx

How can we solve the following trigonometric equation?

in the interval .

Solution

IB Math HL – Trigonometry, Trigonometric equations

First we’ll solve the trigonometric equation

and then the other trigonometric equation .

Therefore, the solutions are

### 12. IB Mathematics HL – Trigonometry, Trigonometric identities, double angle formula

How can we simplify the following trigonometric expression?

Solution

IB Math HL – Trigonometry, Trigonometric identities, double angle formula

We’ll use the following double angle formulas:

and

So, the expression can be written as