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:
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 true for this triangle:
For each of the following triangles, write an equation relating the sides and angles using the sine rule:
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}
Consider the following diagram:
Find an expression for \sin A in \triangle ACD.
Find an expression for \sin B in \triangle BDC. Then make x the subject of the equation.
Substitute your expression for x into the equation from part (a), and rearrange the equation to form the sine rule.
Consider the given triangle:
Write out the sine rule to find h.
Find the value of h to two decimal places.
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:
Calculate the length of y in metres, using the sine rule.
Round your answer correct to one decimal place.
For each of the following triangles:
Find the value of a using the sine rule. Round your answer 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. Round your answer to two decimal places.
Consider the following triangle:
Find the length of side HK to two decimal places.
Find the length of side KJ to two decimal places.
Consider the triangle with \angle C = 72.53 \degree and \angle B = 31.69 \degree, and one side length a = 5.816\text{ m}.
Find \angle A. Round your answer to two decimal places.
Find the length of side b. Round your answer to three decimal places.
Find the length of side c. Round your answer to three decimal places.
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 two decimal places where necessary:
\angle A.
a
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 \triangle QUV where the side lengths q, u and v appear opposite \angle Q, \angle U and \angle V. If q = 16, \sin V = 0.5 and \sin Q = 0.8, find the value of v.
For each of the following diagrams, find the value of the angle x using the sine rule. Round your answers to one decimal place.
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}
The angle of depression from J to M is 68 \degree. The length of JK is 25 \text{ m} and the length of MK is 28 \text{ m} as shown:
Find the following, rounding your answers to two decimal places:
Find x, the size of \angle JMK.
Find the angle of elevation from M to K.
Consider \triangle ABC below:
Find x to the nearest degree.
Find \angle ADB to the nearest degree.
Consider the following diagram of a quadrilateral:
Find the value of \theta, correct to two decimal places.
For each of the following triangles, determine whether the angle x is bigger or smaller than the given angle:
Rochelle needs to determine whether a triangle with the dimensions shown below is possible or not:
Find the value of \theta.
Find the value of \dfrac{8.4}{\sin 106 \degree}. Round your answer to four decimal places.
Find the value of \dfrac{4.0}{\sin \theta}. Round your answer to four decimal places.
Hence, is it possible to construct this triangle? Explain your answer.
\triangle ABC is such that \angle CAB = 32 \degree, a = 5 and b = 9. Let the unknown angle opposite the length 9 \text{ cm} be x.
Consider the acute case, and find the size of angle x, to two decimal places.
Consider the obtuse case, and find the size of the obtuse angle x, to two decimal places.
Determine the number of possible triangles given the following:
a = 39, b = 32, and B = 50 \degree
State whether the following sets of data determine a unique triangle:
a = 5, b = 6, C = 80 \degree
A = 50 \degree, B = 30 \degree, c = 8
A = 20 \degree, B = 40 \degree, C = 120 \degree
a = 5, b = 12, c = 13
B = 40 \degree, b = 2, c = 5
a = 6, b = 3, c = 27
a = 3, b = 4, c = 5
A = 80 \degree, B = 20 \degree, C = 80 \degree
Determine if solving \triangle ABC could result in the ambiguous case given the following:
a, b and c are known.
A, B and a are known.
a, B and c are known.
For each of the following sets of measurements, determine whether \triangle ABC could be acute, obtuse and/or right-angled:
\angle CAB = 36 \degree, a = 7 and b = 10
\angle CAB = 42 \degree, a = 7 and b = 2
\angle CAB = 25 \degree, a = 4 and b = 7
In \triangle ABC, A = 45\degree and c = 5\text{ mm}.
In what interval does the length of BC lead to two possible triangles that can be formed?
Consider the following triangle which is not drawn to scale:
Calculate the sizes of angle, x, to the nearest degree.
Calculate the sizes of angle y, to the nearest degree.
Hence, calculate the side z, correct to two decimal places.
Two of a triangle's side lengths are 13.5\text{ cm} and 16.8\text{ cm}. The angle opposite the side which is 13.5\text{ cm} long is 48 \degree.
Will the angle opposite the side which is 16.8\text{ cm} long be bigger or smaller than 48 \degree?
Calculate the size of the angle x, opposite the side which is 16.8\text{ cm} long, correct to one decimal place.
A line joining the origin and the point \left(6, 8\right) has been graphed on the number plane. To form a triangle with the x-axis, a second line is drawn from the point \left(6, 8\right) to the positive side of the x-axis.
In what interval can the length of the second line be such that there are two possible triangles that can be formed with that length?
What can the length of the second line be such that there is exactly one triangle that can be formed with that length?
For what lengths of the second line will no triangle be formed?
A line joining the origin and the point \left( - 8 , 6\right) has been graphed on the number plane. To form a triangle with the x-axis, a second line is drawn from the point \left( - 8 , 6\right) to the negative x-axis.
In what interval should the length of the new line be such that there are two possible triangles that can be formed with that length?
What can the length of the second line be such that there is exactly one triangle that can be formed with that length?
For what lengths of the second line will no triangle be formed?
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.
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.
Consider the following diagram:
Find the size of \angle OBA.
Find the length of k, to two decimal places.
