IB Maths HL – Trigonometry
Questions and Answers
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