Q: Given $$\triangle ABC$$ with $$a = 8$$, $$b = 6$$, and $$C = 45^\circ$$, what is the length of side $$c$$?

$$c = \sqrt{8^2$ + 6^2$ - 2 \times 8 \times 6 \times \cos 45^\circ}$$

Q: A ship sails 10 km north, then turns $$60^\circ$$ east and sails another 15 km. How far is the ship from its starting point?

$$\sqrt{10^2$ + 15^2$ - 2 \times 10 \times 15 \times \cos 60^\circ}$$

Q: Which of the following identities is correct?

$$\sin^2$ \theta + \cos^2$ \theta = 1$$

Q: A triangle has sides $$a = 7$$, $$b = 9$$, and angle $$C = 120^\circ. $$What is the area of the triangle?

$$\frac{1}{2} \times 7 \times 9 \times \sin 120^\circ$$

Q: A triangle has sides $$a = 5$$, $$b = 7$$, and $$c = 8. $$What is the measure of angle $$C$$?

$$\cos$$^{-1}$$\left(\frac{5^2$ + 7^2$ - 8^2$}{2 \times 5 \times 7}\right)$$

Question 1

A ladder leans against a wall, forming an angle of $$65^\circ$$ with the ground. If the foot of the ladder is 3 meters from the wall, how long is the ladder? (Round your answer to two decimal places.)

Accepted Answer

$$7.13$$ meters

Question 2

Which of the following is the correct expression for the sine rule?

Accepted Answer

$$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$

Question 3

A surveyor measures the angle of elevation to the top of a building as $$30^\circ$$ from a point 50 meters away from the base. How tall is the building?

Accepted Answer

$$28.87$$ meters

Question 4

Given $$	riangle ABC$$ with $$a = 8$$, $$b = 6$$, and $$C = 45^\circ$$, what is the length of side $$c$$?

Accepted Answer

$$c = \sqrt{8^2$ + 6^2$ - 2 	imes 8 	imes 6 	imes \cos 45^\circ}$$

Question 5

Which of the following is the correct formula for the area of a triangle given two sides and the included angle?

Accepted Answer

$$\frac{1}{2}ab\sin C$$

Question 6

A ship sails 10 km north, then turns $$60^\circ$$ east and sails another 15 km. How far is the ship from its starting point?

Accepted Answer

$$\sqrt{10^2$ + 15^2$ - 2 	imes 10 	imes 15 	imes \cos 60^\circ}$$

Question 7

If $$\sin 	heta = 0.6$$ and $$	heta is in $$the first quadrant, what is $$\cos 	heta$$?

Accepted Answer

$$0.8$$

Question 8

Which of the following identities is correct?

Accepted Answer

$$\sin^2$ 	heta + \cos^2$ 	heta = 1$$

Question 9

A ramp rises 2 meters over a horizontal distance of 8 meters. What is the angle of elevation of the ramp?

Accepted Answer

$$\arctan\left(\frac{2}{8}\right)$$

Question 10

If $$	an 	heta = 3$$ and $$	heta is in $$the first quadrant, what is $$\sin 	heta$$?

Accepted Answer

$$\frac{3}{\sqrt{10}}$$

Question 11

A triangle has sides $$a = 7$$, $$b = 9$$, and angle $$C = 120^\circ. $$What is the area of the triangle?

Accepted Answer

$$\frac{1}{2} 	imes 7 	imes 9 	imes \sin 120^\circ$$

Question 12

Which of the following is the correct expression for the cosine rule?

Accepted Answer

$$c^2$$ = $$a^2$$ + $$b^2 - 2ab$$\cos C$$

Question 13

A radio tower is supported by a wire that makes a $$75^\circ$$ angle with the ground and is attached to the top of the tower 20 meters high. How long is the wire?

Accepted Answer

$$\frac{20}{\sin 75^\circ}$$

Question 14

If $$\sin \alpha = 0.5$$ and $$\alpha is in $$the second quadrant, what is $$\alpha$$?

Accepted Answer

$$150^\circ$$

Question 15

A cyclist rides along a straight road that rises at an angle of $$5^\circ to $$the horizontal. If the cyclist travels 2 km along the road, what is the vertical height gained?

Accepted Answer

$$2 	imes \sin 5^\circ$$

Question 16

Which of the following is the correct value for $$\sin 60^\circ$$?

Accepted Answer

$$\frac{\sqrt{3}}{2}$$

Question 17

A triangle has sides $$a = 5$$, $$b = 7$$, and $$c = 8. $$What is the measure of angle $$C$$?

Accepted Answer

$$\cos$$^{-1}$$\left(\frac{5^2$ + 7^2$ - 8^2$}{2 	imes 5 	imes 7}ight)$$

Question 18

If $$\cos 	heta = -0.4$$ and $$	heta is in $$the second quadrant, what is $$	heta$$?

Accepted Answer

$$\cos$$^{-1}$$(-0.4)$$

Question 19

A building casts a shadow 12 meters long when the angle of elevation of the sun is $$30^\circ. $$How tall is the building?

Accepted Answer

$$12 	imes 	an 30^\circ$$

Trigonometry Study Notes: NCEA Level 2 Internal (Walker Maths 2.4)

Guida di studio - Domande di pratica

Study guide - Flashcard