# 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    