For each of the given triangles, determine if there is enough information to find all the remaining sides and angles in the triangle using only the sine rule:
Three sides are known:
Two of the angles and the side included between them are known:
Two of the angles and a side not included between them are known:
Two of the sides and an angle included between them are known:
For each of the following triangles, write an equation relating the sides and angles using the sine rule:
An oblique \triangle ABC consists of angles A, B and C which appear opposite sides a, b and c respectively. State whether the following equations are correct:
\dfrac{\sin B}{\sin C} = \dfrac{b}{c}
\dfrac{a}{\sin A} = \dfrac{c}{\sin C}
\dfrac{a}{\sin A} = \dfrac{\sin C}{c}
\dfrac{\sin A}{a} = \dfrac{\sin B}{c}
For the given triangle, use the sine rule to find the value of h to two decimal places:
Solve the following equations for x, given that all equations relate to acute-angled triangles. Round your answers to two decimal places.
\dfrac{x}{\sin 78 \degree} = \dfrac{50}{\sin 43 \degree}
\dfrac{11}{\sin 66 \degree} = \dfrac{x}{\sin 34 \degree}
\dfrac{\sin 78 \degree}{20} = \dfrac{\sin x}{13}
\dfrac{5}{\sin x} = \dfrac{7}{\sin 70 \degree}
\dfrac{22}{\sin x} = \dfrac{31}{\sin 66 \degree}
Complete the following steps in order to prove the sine rule:
Consider the \triangle ACD, find an expression for \sin A.
Consider \triangle BDC, find an expression for \sin B.
Make x the subject of the equation from part (b).
Substitute your expression for x into the equation from part (a) to prove the sine rule.
For each of the following triangles, find the side length a using the sine rule. Round your answers to two decimal places.
For each of the following triangles, find the length of side x, correct to one decimal place:
Consider the following triangle:
Find the length a using the sine rule. Give your answer correct to two decimal places.
Use another trigonometric ratio and the fact that the triangle is right-angled to calculate and confirm the value of a. Give your answer correct to two decimal places.
For each of the folowing right-angled triangles:
Find the length a, using the sine rule.
Use the trigonometric ratios and the fact that the triangle is right-angled to calculate and confirm the value of a.
Consider the following triangle:
Find the length of side HK. Round your answer to two decimal places.
Find the length of side KJ. Round your answer to two decimal places.
Calculate the length of y in metres, using the sine rule.
Round your answer correct to one decimal place.
Consider the triangle \triangle QUV where the side lengths q, u and v appear opposite the angles Q, U and V. If q = 16, \sin V = 0.5 and \sin Q = 0.8, then solve for v.
For each of the following diagrams, find the value of the acute angle x using the sine rule. Round your answers to one decimal place.
Find the value of the acute angle x.
Round your answer to two decimal places.
For each of the following acute angled triangles, calculate the size of angle B to the nearest degree:
\triangle ABC where \angle A = 57 \degree side a = 156 \text{ cm} and side b = 179 \text{ cm}
\triangle ABC where \angle A = 48 \degree side a = 2.7 \text{ cm} and side b = 1.9 \text{ cm}
Consider the following diagram of a quadrilateral:
Find the value of \theta, correct to two decimal places.
Consider the given triangle:
Find the size of \angle BAC.
Find the value of c, correct to two decimal places.
Consider the triangle with \angle B = 58 \degree, \angle C = 29 \degree and a = 36 \text{ cm}. Find the following, rounding your answers to two decimal places where necessary:
\angle A
b
c
Consider the triangle with \angle B = 82.94 \degree, \angle C = 60.25 \degree, and a side length of c = 19.84 \text{ cm}. Find the following, rounding your answers to two decimal places where necessary:
\angle A
a
b
Consider the triangle with \angle B = 38.18 \degree, \angle C = 81.77 \degree and b = 54 \text{ m}. Find the following, rounding your answers to the nearest whole number:
\angle A
a
c
Consider the triangle where \angle A = 34 \degree, \angle C = 91 \degree and c = 15 \text{ cm}. Find the following, rounding your answers to two decimal places where necessary:
\angle B
a
b
Consider the given triangle with two interior angles C = 72.53 \degree and B = 31.69 \degree, and one side length a = 5.816 \text{ m}.
Find the size of angle A, to two decimal places.
Find the value of b, to three decimal places.
Find the value of c, to three decimal places.
A radio signal is sent from a transmitter at tower T, via a satellite S, to a town W, as shown in the diagram. The town is 526 \text{ km} from the transmitter tower. The signal is sent out from the transmitter tower at an angle of 18 \degree, and the town receives the signal at an angle of 26 \degree.
Find the size of \angle WST.
Find the distance, SW, that the signal travels from the satellite to the town. Round your answer to the nearest kilometre.
If the satellite is h \text{ km} above the ground, find h. Round your answer to two decimal places.
Consider the following diagram:
Find the size of \angle OBA.
Find the length of k, to two decimal places.
Dave is standing on a hill and can see two buildings in the distance. Suppose the buildings are 20 \text{ km} apart. Dave is 13 \text{ km} from one building and the angle between the two lines of sight to the buildings is 35 \degree.
Find the size of \angle ABC, correct to two decimal places.
Find the size of \angle BCA, correct to two decimal places.
Find the distance from Dave to the building at B, correct to one decimal place.
A bridge connects two towns on either side of a gorge, where one side of the gorge is inclined at 59 \degree and the other side is inclined at 70 \degree. The length of the steeper incline is 59.1 \text{ m}.
Find x, the length of the bridge. Round your answer correct to one decimal place.
During football training, the coach marks out the perimeter of a triangular course that players need to run around. The diagram shows some measurements taken of the course, where side length a = 14 \text{ m}:
Find the size of \angle A.
Find the length of side c, correct to two decimal places.
Find the length of side b, correct to two decimal places.
Each player must sprint one lap and then jog one lap around the triangle. This process is to be repeated 3 times by each player.
If Tara can run 280 \text{ m/min}, and can jog at half the speed she runs, calculate the time this exercise will take her, correct to one decimal place.
Two wires help support a tall pole. One wire forms an angle of 36 \degree with the ground and the other wire forms an angle of 70 \degree with the ground. The wires are 29 \text{ m} apart.
Find a, the angle made between the two wires at the top of the pole.
Find d, the length of the longest wire in metres. Round your answer to two decimal places.
Find h, the height of the pole. Round your answer to two decimal places.
Mae observes a tower at an angle of elevation of 12 \degree. The tower is perpendicular to the ground. Walking 67 \text{ m} towards the tower, she finds that the angle of elevation increases to 35 \degree:
Calculate the size of \angle ADB.
Find the length of the side a. Round your answer to two decimal places.
Find the height, h, of the tower. Round your answer to one decimal place.
To calculate the height of each block of flats, a surveyor measures the angles of depression from A and B, to C. From A the angle of depression is 31 \degree, and from B the angle of depression is 47 \degree.
Find the size of \angle ACB.
If the distance between A and C is b \text{ m}, find the value of b. Round your answer to two decimal places.
If the buildings are h \text{ m} tall, find the value of h. Round your answer to the nearest metre.
