Topic 4: Right triangles and trigonometry

Label the sides opposite, adjacent and hypotenuse from the angle you are using. Check the calculator is in degree mode before you use sine, cosine or tangent.

A lesson for this topic

Part I style: multiple choice

Question 1Part I style

In right triangle

ABC

\angle C

is a right angle,

AC

= 24

BC

= 7

and

AB

= 25

. What is

\sin A

Question 2Part I style

\sin 9^\circ

= \cos x^\circ

and

0

< x

< 90

, what is the value of

x

Question 3Part I style

\sin(3x + 4)^\circ

= \cos(5x - 2)^\circ

and both angles are acute, what is the value of

x

Question 4Part I style

Two different right triangles each have an acute angle of

40^\circ

. Why is the ratio of the side opposite the

40^\circ

angle to the hypotenuse the same in both triangles?

Question 5Part I style

In right triangle

ABC

\angle C

is a right angle. Which equation is always true?

Question 6Part I style

In right triangle

ABC

\angle C

is a right angle and

\cos A

= \frac{5}{13}

. What is

\tan A

Question 7Part I style

A right triangle has a hypotenuse of length 17 and one leg of length 8. What is the length of the other leg?

Question 8Part I style

In right triangle

ABC

\angle C

is a right angle,

m\angle A

= 38^\circ

and

AB

= 14

. What is

BC

, to the nearest tenth?

Question 9Part I style

In right triangle

ABC

\angle C

is a right angle,

BC

= 9

and

AC

= 14

. What is

m\angle A

, to the nearest degree?

Question 10Part I style

A 25-foot ladder leans against a vertical wall on level ground. The ladder makes an angle of

76^\circ

with the ground. How high up the wall does the ladder reach, to the nearest tenth of a foot?

Question 11Part I style

From a point on level ground 26 feet from the base of a tree, the angle of elevation to the top of the tree is

75^\circ

. What is the height of the tree, to the nearest tenth of a foot?

Question 12Part I style

From the top of an 85-foot lighthouse, the angle of depression to a boat is

12^\circ

. The base of the lighthouse is at the same level as the boat. How far is the boat from the base of the lighthouse, to the nearest tenth of a foot?

Question 13Part I style

In a

30^\circ

60^\circ

90^\circ

triangle, the hypotenuse is 14 units long. What is the length of the longer leg?

Question 14Part I style

The diagonal of a square is 10 inches long. What is the length of a side of the square, in inches?

Question 15Part I style

A loading ramp rises 4 feet over a horizontal distance of 9 feet. What angle does the ramp make with the ground, to the nearest tenth of a degree?

Question 16Part I style

Rosa's eyes are 5 feet above level ground. She stands 140 feet from a building and measures the angle of elevation to its top as

35^\circ

. How tall is the building, to the nearest tenth of a foot?

Question 17Part I style

Which set of lengths could be the sides of a right triangle?

Question 18Part I style

In right triangle

ABC

\angle C

is a right angle,

m\angle A

= 12^\circ

and

AC

= 30

. What is

AB

, to the nearest tenth?

Question 19Part I style

\triangle PQR

PQ

= 14

QR

= 19

and

m\angle Q

= 40^\circ

. What is the area of the triangle, to the nearest tenth?

Question 20Part I style

\triangle ABC

, the lengths

AB

and

BC

and the measure of

\angle B

are known. Which expression gives the area of the triangle?

Question 21Part I style

In acute

\triangle ABC

\overline{AD}

is the altitude from

A

\overline{BC}

. Which expression is equal to

AD

Question 22Part I style

A triangular garden has an area of 60 square meters. Two of its sides meet at a

30^\circ

angle; one of them is 12 meters long. How long is the other side, in meters?

Constructed response: Parts II, III and IV

Sine and cosine of complementary anglesPart II style · 2 credits

In a right triangle, the acute angles measure

(x + 20)^\circ

and

(2x + 10)^\circ

, and

\sin(x + 20)^\circ

= \cos(2x + 10)^\circ

(a) Find the value of

x

. [1]

Type fractions as 3/8, roots as sqrt(5), pi as pi (so 12pi), points as (3, -5) and equations as y=2x+1.

(b) Explain why the sine of one acute angle of a right triangle equals the cosine of the other acute angle. [1]

0 words · a sentence

The area of a triangular gardenPart II style · 2 credits

A garden is shaped like

\triangle ABC

, with

AB

= 9

feet,

AC

= 14

feet and

m\angle A

= 70^\circ

. Segment

\overline{BD}

is perpendicular to

\overline{AC}

, and its length is

h

(a) Find

h

, to the nearest tenth of a foot. [1]

Type fractions as 3/8, roots as sqrt(5), pi as pi (so 12pi), points as (3, -5) and equations as y=2x+1.

(b) Find the area of the garden, to the nearest tenth of a square foot. [1]

Type fractions as 3/8, roots as sqrt(5), pi as pi (so 12pi), points as (3, -5) and equations as y=2x+1.

A tower on a cliffPart III style · 4 credits

Kayla stands at point

A

on level ground, 200 feet from point

B

at the base of a vertical cliff. A vertical tower

\overline{CT}

stands on top of the cliff, directly above

B

. From

A

, the angle of elevation to

C

, the top of the cliff, is

24^\circ

, and the angle of elevation to

T

, the top of the tower, is

31^\circ

(a) Find the height of the cliff,

BC

, to the nearest tenth of a foot. [1]

Type fractions as 3/8, roots as sqrt(5), pi as pi (so 12pi), points as (3, -5) and equations as y=2x+1.

(b) Find

BT

, the height of the top of the tower above the ground, to the nearest tenth of a foot. [1]

Type fractions as 3/8, roots as sqrt(5), pi as pi (so 12pi), points as (3, -5) and equations as y=2x+1.

CT

, to the nearest foot. [1]

Type fractions as 3/8, roots as sqrt(5), pi as pi (so 12pi), points as (3, -5) and equations as y=2x+1.

(d) Find the distance

AT

from Kayla to the top of the tower, to the nearest foot. [1]

Type fractions as 3/8, roots as sqrt(5), pi as pi (so 12pi), points as (3, -5) and equations as y=2x+1.