Trigonometry : 9th grade math worksheets with answers

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9th grade math worksheets with answers on trigonometry in the right triangle.

All these exercises in third grade have their answers and you can download them in PDF format.

Exercise 1 – Perimeter
we know that \widehat{CAB}=50^{\circ}; \widehat{DBA}=15^{\circ}; \widehat{ACB}=90^{\circ} and AB=40\,m.
Calculate the perimeter of triangle ABD. Round the result to the nearest decimeter.

Trigonometry in the right triangle

Exercise 2 – Cable
A 20m long cable is stretched between the top of a vertical pole and the horizontal ground.
It forms an angle of 40° with the ground.
a) Calculate the height of the pole; give the value approximated to the nearest tenth by default.
b) Represent the situation by the figure at 1/200 scale. (The data for the situation should be placed on the figure).

Exercise 3 – Calculator
1.a. Using a calculator, calculate (cos67°+sin67°)²+(cos67°-sin67°)² (cos35°+sin35°)²+(cos35°-sin35°)² b. What do we see?

Prove that for any acute angle x : (cos x+sin x)²+(cos x-sin x)²= 2.

Exercise 4 – Right-angled triangle and trigonometry
Show that the triangle SON is a rectangle.

Trigonometry in the right triangle

Exercise 5 – Calculation of the tangent and cosine
x is an angle such that sinx=\frac{5}{8}.

Using the trigonometric formulas, calculate the exact values of cos\,x and tan\,x.

Exercise 6 – Circumscribed circle
1. Construct a triangle ABC at C such that AC = 5 cm and \widehat{BAC}=40^{\circ}.

2. Calculate the length BC. (A value rounded to the nearest millimeter will be given).

3.a) Where is the center O of the circumscribed circle of triangle ABC?

b) Draw this circle.

4. Deduce the measure of the angle \widehat{BOC}.

Exercise 7 – Measuring the height of a cathedral
This cathedral is 140 m high.

The camera is equipped with a lens with an opening angle of 42°.
height of a cathedral
What is the OH distance needed for the cathedral to appear fully in the lens?

Exercise 8 – Trigonometry in the right triangle.

We know that AB = 40 m, \widehat{A}=45^{\circ} and \widehat{B}=25^{\circ}.

We are looking for the SH height of the keep.

Let x = Sh (with x expressed in m).

a. Express BH as a function of x.

b. Express AH as a function of x.

c. Knowing that AB = 40 m, write an equation whose unknown is x.

d. Solve the equation and give the height of the dungeon, rounding to the nearest meter.

Mont Saint-Michel

The series 3 of the exercises on trigonometry in the right-angled triangle in 3ème. These exercises, corrected in detail, on cosine, sine and tangent of an angle are available for free download in PDF in the third grade.

Exercise 9 – Cube and trigonometry
right block

In the right block above, we give EH=69 cm, EF=60 cm and EA=51 cm.
What is the measure of the angle AED? (round the result to the unit)

Exercise 10 – Mont-Saint-Michel and trigonometry
To find out how high the archangel Saint Michael reaches

located at the top of the abbey of Mont-Saint-Michel, Lisa uses a theodolite

to measure angles from the bay.

Mont Saint-Michel and trigonometry

She makes a first measurement and observes the top of the abbey under an angle of 48°.

Then she moves back 50 m and takes another measurement.

She then sees the top of the abbey at an angle of 40°.

Her father tells her that she can now find the height of the top of the abbey.

Help Lisa do this calculation with the help of the diagram below:

Trigonometry exercise

Exercise 11 – Al-kashi formula
1. Construct a full-size triangle ABC such that: AB = 7 cm; BC = 8 cm and AC = 5 cm.

2. Is the triangle ABC rectangular ? Justify.

3. The famous mathematician Al-Kashi (XIV- XV century) found a formula between the three
sides of a triangle and the cosine of one of the three angles of this triangle :

“In a triangle, whether it is right-angled or not, we have :
BC^2\,=\,AB^2\,+\,AC^2\,-\,2\,\times  \,AB\,\times  \,AC\,\times  \,cos\,\,\widehat{BAC}. ”

Using Al-Kashi’s formula, find the measure rounded to the nearest degree of the angle \widehat{BAC}.

Exercise 12 – Slope of a road and highway
A slope of 15% means that, for a horizontal displacement, we move vertically of 15 m as shown below.

Slope of a road

1) For a 15% slope, what angle does the road make with the horizontal?

2) A dangerous descent is considered as soon as the slope is higher than 10% on the road and higher than 4% on the highway.

From what angle between the road and the horizontal, is it considered dangerous to go downhill on a road?

3) Is it more dangerous to drive on a road with a 20% slope or to drive on a highway with a 20 degree angle to the horizontal? Justify

Percentage of a slope

The series 4 of the exercises on trigonometry in 3ème with the cos, sin and tan of an acute angle in a right triangle. These exercises in the third grade have their detailed correction and can be downloaded and printed in PDF.

Exercise 13

1. Construct a triangle IJK such that :

JK = 8 cm; IJ = 4.8 cm; KI = 6.4 cm.

2. Prove that the triangle IJK is a right triangle.

3. Calculate the measure in degrees of the angle \,\widehat{IJK}.

Give the value rounded to the nearest degree.

Exercise 14

1. Paul wants to install a basketball hoop at his house.

It must be fixed at 3.05 m from the ground.

The ladder he uses is 3.20 m long.

How far from the foot of the wall should he place the ladder so that its top is just level with the basket?

(Give an approximate value to the nearest cm).

2. Calculate the angle formed by the ladder and the ground. (Give a value approximated to the nearest degree).

Basketball basket

Exercise 15

Let ABC be an isosceles triangle with base [BC], [AH] the height from vertex A.

We have: BC = 8 cm and AH = 7 cm.

1. Construct the triangle ABC and justify the construction.

2. Calculate tan(B) .

3. Deduce the value of the angle \widehat{B} rounded to the nearest degree.

Exercise 16

ABCD is a rectangle such that AB = 7.2 cm and BC = 5.4 cm.

1. Draw this rectangle and its diagonal in real size [AC].

2. Calculate the measure rounded to the degree of the angle \,\widehat{ACD}.

3. Show that the angles \,\widehat{ACD} and \,\widehat{CAB} are equal.

4. The bisector of the segment [AC] intersects the line (AB) at E.

Place the point E and show that the triangle ACE is isosceles.

5. Deduce an approximate value for the measure of the angle \,\widehat{DCE}.

Exercise 17

The unit of length is the centimeter.


The rectangle above represents a pool table.

Two billiard balls N and B are placed such that: CD = 90; NC = 25; BD = 35.

(The angles are right.)

A player wants to hit the N-ball with the B-ball along the path BEN, with E between C and D, and such that the measure of the angle \,\widehat{CEN} is equal to that of \,\widehat{DEB}.

. Let ED = x.


a. Give a frame for x.

b. Express CE as a function of x.

2. In the triangle BED, express \,tan(\widehat{DEB}) as a function of x.

3. In the NEC triangle, express \,tan(\widehat{CEN}) as a function of x.


a. By equating the two quotients found in question 2. and 3.,

we find the equation: 35(90 – x) = 25 x.

We don’t ask you to justify it.

Solve this equation.

b. Deduce the common value of the angles \,\widehat{CEN} and \,\widehat{DEB} rounded to the degree.

Exercise 18: Calculating the measure of an angle
Calculate, for each figure, the measure of the marked angle (round the result to the nearest degree).

Trigonometry in the right triangle

Exercise 19: Vendée sailboats
The Vendée and Globe sailboats are off an island and will join it to spend the night.
We can schematize their positions V and G as shown opposite.

Knowing that the sailboat Vendée is 800 m from the island, determine to the nearest meter the distance separating Globe from the island and the distance between the two sailboats.

Vendée of globes and trigonometry

Exercise 20: Crossed triangles
1. Calculate the measure of \widehat{IGH}.

2. Deduce the measure of the angle \widehat{EGF}.

3. Calculate the lengths EF and FG rounded to the tenth.


Exercise 21: trigonometry with two triangles
Calculate the length OM rounded to the millimeter.

Trigonometry in the right triangle

Exercise 22: Trigonometry in the right triangle
We give BD = 4 cm , BA = 6 cm and \widehat{DBC}=60^{\circ}.

1. Show that BC= 8 cm.

2. Calculate CD.

Give the value rounded to the tenth.

3. Calculate AC.

4. What is the value of tan\widehat{BAC}?

5. Deduce the value, rounded to the nearest degree, of \widehat{BAC} .

Trigonometry in the right triangle

Corrected math exercises.

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