11.06 Sine rule

For each of the following, determine if there is enough information to solve for the remaining sides and angles using the sine rule:

a

A triangle where two of the angles and the side included between them are known.

b

A triangle where two of the angles and a side not included between them are known.

c

A triangle where two of the sides and an angle included between them are known.

d

Consider a triangle where all three sides are known, but no angles are known.

An oblique \triangle ABC consists of angles A, B and C which appear opposite sides a, b and c respectively.

Determine if the following are true or false:

Consider the following parallelogram:

a

Find the area of the triangle formed by the points X, Y and Z, in terms of x, y, and \sin Z.

b

Hence, find the area of the parallelogram in terms of x, y, and \sin Z.

Consider the following diagram:

a

Find an expression for \sin A in \triangle ACD.

b

Find an expression for \sin B in \triangle BDC. Then make x the subject of the equation.

c

Substitute your expression for x into the equation from part (a), and rearrange the equation to form the sine rule.

Find the length of side a in the following triangles, rounding your answers to two decimal places:

a

b

Find the length of side a in the triangles below, correct to two decimal places:

a

b

For the following triangles, find the length of side x correct to one decimal place:

a

b

Consider the triangle \triangle QUV whose side lengths q, u and v are opposite the angles Q, U and V.

Find the value of v if q = 16, \sin V = 0.5 and \sin Q = 0.8.

Find the size of the acute angle x in the following triangles, correct to one decimal place:

a

b

Find the size of angle x in the following triangles, correct to two decimal places:

a

b

c

Consider the triangle where A = 44 \degree, C = 95 \degree and c = 29\text{ m}

Consider the triangle where B = 58 \degree, C = 29 \degree and a = 36\text{ m}.

Consider the triangle with B = 38.18 \degree, C = 81.77 \degree and b = 54\text{ m}.

Consider the triangle with angles A, B = 82.94 \degree and C = 60.25 \degree. The triangle has unknown side lengths a and b, and a third side length c = 19.84 centimetres.

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.

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.

Rochelle needs to determine whether a triangle with the dimensions shown below is possible or not:

a

Find the value of \theta.

b

Find the value of \dfrac{8.4}{\sin 106 \degree}. Round your answer to four decimal places.

c

Find the value of \dfrac{4.0}{\sin 26 \degree}. Round your answer correct to four decimal places.

d

Hence, is it possible to construct this triangle? Explain your answer.

Consider \triangle ABC below:

a

Find x, given that x is acute. Round your answer to the nearest degree.

b

Find \angle ADB to the nearest degree, given that \angle ADB > x.

\triangle ABC consists of angles A, B and C which appear opposite sides a, b and c respectively.

Determine if solving \triangle ABC could result in the ambiguous case given the following:

Determine the number of possible triangles given the following:

Determine whether the following sets of data determine a unique triangle:

For each of the given measurements of \triangle ABC:

\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.

In \triangle ABC, A = 45\degree and c = 5\text{ mm}.

What is the range of lengths, rounded to the nearest tenth where appropriate, for BC that lead to the ambiguous case where we don't know if the triangle formed is acute or obtuse?

Two wires help support a tall pole. One wire forms an angle of 36 \degreewith the ground and the other wire forms an angle of 70 \degreewith the ground. The wires are 29 \text{ m} apart:

A radio signal is sent from a transmitter at tower T, via a satellite S, to a town W, as shown in the diagram below. 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.

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.

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 \degreeand from B the angle of depression is 47 \degree.

a

Find the size of \angle ACB.

b

If the distance between A and C is b \text{ m}, find the value of b. Round your answer to two decimal places.

c

If the buildings are h \text{ m} tall, find the value of h. Round your answer to the nearest metre.

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:

a

Find the value of \angle A.

b

Find the length of side c. Round your answer to two decimal places.

c

Find the length of side b. Round your answer to two decimal places.

d

Each player is to sprint one lap and then jog one lap alternately, doing each 3 times.

If Tara can run 280 \text{ m/min}, and can jog at half the speed she runs, how long will this exercise take her? Round your answer to one decimal place.