Trigonometry – Year 12 A level maths worksheet with answers to download as PDF

In triangle \(ABC\), angle \(A=48^\circ\), angle \(B=65^\circ\) and side \(a=BC=12\) cm.

1. Find angle \(C\). [1 mark]
2. Use the sine rule to find the length \(b\) to 3 significant figures. [3 marks]
3. Find the length \(c\) to 3 significant figures. [3 marks]

Use the cosine rule and area formula.

1. In triangle \(ABC\), \(a=7\) cm, \(b=9\) cm and angle \(C=52^\circ\). Find \(c\) to 3 significant figures. [3 marks]
2. Triangle \(PQR\) has sides 8 cm, 11 cm and 14 cm. Find the smallest angle to 1 decimal place. [4 marks]
3. A triangle has sides 6.4 cm and 9.1 cm with an included angle of \(38^\circ\). Find its area to 3 significant figures. [2 marks]
4. A triangle with sides 8 cm and 10 cm has area 20 cm\(^2\). Find the two possible sizes of the included angle. [3 marks]

Answer these questions using radians.

1. Convert \(135^\circ\) to radians and \(\dfrac{5\pi}6\) to degrees. [2 marks]
2. A sector has radius 8 cm and angle 1.2 radians. Find the arc length and the area of the sector. [3 marks]
3. A sector has radius 6 cm and arc length 15 cm. Find its angle and its area. [3 marks]
4. A circle has radius 10 cm. A chord subtends an angle of 1.2 radians at the centre. Find the area of the segment to 2 decimal places. [3 marks]

Without a calculator, find the exact value of each expression.

1. \(\sin60^\circ\cos30^\circ-\tan45^\circ\) [3 marks]
2. \(\dfrac{\tan\frac\pi3}{\sin\frac\pi4}\) [3 marks]
3. \(\cos150^\circ\), \(\sin210^\circ\) and \(\tan135^\circ\) [3 marks]

Use \(\tan\theta\equiv\dfrac{\sin\theta}{\cos\theta}\) and \(\sin^2\theta+\cos^2\theta\equiv1\).

1. Prove that \(\dfrac{1-\cos^2\theta}{\sin\theta\cos\theta}\equiv\tan\theta\). [3 marks]
2. Prove that \(\tan\theta+\dfrac1{\tan\theta}\equiv\dfrac1{\sin\theta\cos\theta}\). [4 marks]
3. Given that \(\sin\theta=\dfrac35\) and \(\theta\) is acute, find the exact values of \(\cos\theta\) and \(\tan\theta\). [3 marks]

Solve each equation for \(0^\circ\le\theta\le360^\circ\). Give answers to 1 decimal place where necessary.

1. \(\sin\theta=0.6\) [2 marks]
2. \(2\cos\theta=-1\) [2 marks]
3. \(\tan\theta=-2\) [2 marks]
4. \(\sin2\theta=0.5\) [3 marks]

A surveyor stands at a point \(O\) on one side of a lake. The points \(A\) and \(B\) are on opposite shores. \(OA=120\) m, \(OB=95\) m and angle \(AOB=70^\circ\).

1. Find the width \(AB\) of the lake to the nearest metre. [3 marks]
2. Find the area of triangle \(OAB\) to the nearest m\(^2\). [2 marks]
3. The triangle of land \(OAB\) is for sale at £14.50 per m\(^2\). Work out the price to the nearest £100. [2 marks]
4. Find angle \(OAB\) to 1 decimal place. [3 marks]

A ship leaves port \(P\) and sails 20 km on a bearing of \(040^\circ\) to a buoy \(Q\). It then sails 30 km on a bearing of \(110^\circ\) to a lighthouse \(R\).

1. Show that angle \(PQR=110^\circ\). [2 marks]
2. Find the distance \(PR\) to 1 decimal place. [3 marks]
3. Find the bearing of \(R\) from \(P\) to the nearest degree. [4 marks]
4. The ship returns directly from \(R\) to \(P\) at 18 km/h. How long does the return take, to the nearest minute? [2 marks]

Two observers stand on level ground in line with the base of a tall tower. Observer \(A\) is further away. The observers are 40 m apart (\(AB=40\) m). The angle of elevation of the top \(T\) of the tower is \(32^\circ\) from \(A\) and \(41^\circ\) from \(B\).

1. Show that angle \(ATB=9^\circ\). [2 marks]
2. Use the sine rule in triangle \(ATB\) to find \(AT\) to 1 decimal place. [3 marks]
3. Find the height of the tower to the nearest metre. [2 marks]
4. How far is \(B\) from the base of the tower, to the nearest metre? [3 marks]

The depth of water in a harbour entrance is modelled by \(h=4+2\sin(30t^\circ)\) metres, where \(t\) is the number of hours after midnight, \(0\le t\le12\).

1. Find the depth at 3 am and at 9 am. [2 marks]
2. Find the times when the depth is 5 m. [3 marks]
3. A ferry needs at least 4.5 m of water. Between which times, to the nearest minute, can it enter during this period? [4 marks]
4. State the period of the model. Explain what it means. [2 marks]

A garden designer plans a flower bed in the shape of a sector of a circle with radius 6 m and angle \(\dfrac{2\pi}3\) radians. Fencing costs £6.50 per metre and turf costs £4.20 per m\(^2\).

1. Find the area of the bed in terms of \(\pi\) and as a decimal. [3 marks]
2. Find the perimeter (the arc and two straight edges). [3 marks]
3. Work out the cost of fencing all the way round. [2 marks]
4. Work out the cost of turf for the bed. [2 marks]

A window has a circular segment of coloured glass cut from a circle of radius 10 cm. The chord \(AB\) subtends an angle of 1.2 radians at the centre \(O\).

1. Find the length of the chord \(AB\) to 2 decimal places. [3 marks]
2. Find the area of triangle \(OAB\). [2 marks]
3. Find the area of the segment. [3 marks]
4. The glass costs £0.12 per cm\(^2\). Find the cost of the segment. [2 marks]

A triangular field has sides of 120 m, 150 m and 200 m. Seed costs £0.45 per m\(^2\).

1. Find the largest angle of the triangle. [3 marks]
2. Find the area of the field. [2 marks]
3. Check your area using Heron\'s formula. [3 marks]
4. Find the cost of the seed to the nearest £10. [2 marks]

A light show pattern is described by \(y=2\sin^2x+\sin x-1\), where \(x\) is in degrees.

1. Factorise the quadratic expression in \(\sin x\). [2 marks]
2. Solve \(y=0\) for \(0^\circ\le x\le360^\circ\). [4 marks]
3. Find the maximum and minimum values of \(y\). [3 marks]

A plot of land \(ABCD\) is divided by the diagonal \(AC\) into two triangles. The measurements are in the table.

1. Find \(AC\) to 2 decimal places. [3 marks]
2. Find angle \(ADC\) to 1 decimal place. [4 marks]
3. Find the total area of \(ABCD\). [4 marks]
4. The owner wants a fence along \(AC\) costing £11 per metre. Find the cost. [1 mark]

Use identities to solve each problem.

1. Solve \(2\cos^2\theta+5\sin\theta-4=0\) for \(0^\circ\le\theta\le360^\circ\). [5 marks]
2. Solve \(\tan\theta=2\sin\theta\) for \(0^\circ\le\theta\le360^\circ\). [5 marks]
3. Solve \(\cos(\theta+40^\circ)=-\tfrac12\) for \(0^\circ\le\theta\le360^\circ\). [4 marks]

A sector of a circle has perimeter 20 cm. The radius is \(r\) cm and the angle is \(\theta\) radians. Here are some possibilities.

1. Show that \(\theta=\dfrac{20-2r}r\). [2 marks]
2. Check the table for \(r=4\) and \(r=8\). [2 marks]
3. Show that the area is \(A=r(10-r)\). [2 marks]
4. Complete the square to find the greatest possible area and the matching value of \(\theta\). [4 marks]

The graph shows \(y=3\sin2x+1\) for \(0^\circ\le x\le360^\circ\).

1. State the maximum and minimum values of \(y\). [2 marks]
2. State the period. [1 mark]
3. Solve \(3\sin2x+1=2.5\) for \(0^\circ\le x\le360^\circ\). [4 marks]
4. Describe the transformations that map \(y=\sin x\) to \(y=3\sin2x+1\). [3 marks]

Work with identities to solve a problem.

1. Prove that \(\dfrac{\cos\theta}{1-\sin\theta}\equiv\dfrac{1+\sin\theta}{\cos\theta}\). [4 marks]
2. Prove that \(\dfrac1{1+\sin\theta}+\dfrac1{1-\sin\theta}\equiv\dfrac2{\cos^2\theta}\). [3 marks]
3. Hence solve \(\dfrac1{1+\sin\theta}+\dfrac1{1-\sin\theta}=8\) for \(0^\circ\le\theta\le360^\circ\). [4 marks]

In triangle \(ABC\), \(a=7\) cm, \(b=10\) cm and angle \(A=40^\circ\).

1. Show that there are two possible triangles and find the two possible values of angle \(B\). [4 marks]
2. For each triangle find the length \(c\). [4 marks]
3. For fixed \(b=10\) and \(A=40^\circ\), find the range of values of \(a\) for which two triangles are possible. [3 marks]