The table shows some exact trigonometric values. One value is given for you. Complete the table, giving each missing value in its simplest exact form.
(a)cos 30(1)
(b)tan 30(1)
(c)sin 45(1)
(d)cos 45(1)
(e)tan 45(1)
(f)sin 60(1)
(g)cos 60(1)
(h)tan 60(1)
(Total for Question 1 is 8 marks)
2
Write down the exact value of each of the following.
(a)sin 0(1)
(b)cos 0(1)
(c)sin 90(1)
(d)cos 90(1)
(Total for Question 2 is 4 marks)
3
Which of these is the exact value of cos 60?
A) 1/2
B) √3/2
C) √2/2
D) 1
(Total for Question 3 is 1 mark)
4
Work out the exact value of sin 60 + cos 60. Give your answer as a single fraction in its simplest form.
(Total for Question 4 is 2 marks)
5
Work out the exact value of 4 cos 60 - 2 sin 30. Show your working.
(Total for Question 5 is 2 marks)
6
Work out the exact value of tan 60 x tan 30. Show your working clearly.
(Total for Question 6 is 3 marks)
7
Solve, for 0 ≤ x ≤ 90, sin x = √3/2
(Total for Question 7 is 2 marks)
8
Solve, for 0 ≤ x ≤ 90, cos x = √2/2
(Total for Question 8 is 2 marks)
9
Solve, for 0 ≤ x ≤ 90, tan x = √3/3. Give a reason for your answer.
(Total for Question 9 is 2 marks)
10
Triangle ABC has a right angle at B. Angle A = 30 degrees and AC = 8 cm. Calculate the length of BC. Give your answer as an exact value.
Diagram NOT accurately drawn
(Total for Question 10 is 2 marks)
11
Triangle PQR has a right angle at Q. Angle P = 60 degrees and PQ = 5 cm. Calculate the length of QR. Give your answer as an exact value in the form k √3.
Diagram NOT accurately drawn
(Total for Question 11 is 3 marks)
12
Triangle DEF has a right angle at E. Angle D = 45 degrees and EF = 6 cm. Calculate the length of the hypotenuse DF. Give your answer as an exact value in the form k √2.
Diagram NOT accurately drawn
(Total for Question 12 is 3 marks)
13
A right-angled isosceles triangle has two equal sides of length 7 cm, meeting at a right angle. Calculate the length of the hypotenuse. Give your answer as an exact value in the form k √2.
Diagram NOT accurately drawn
(Total for Question 13 is 2 marks)
14
A ramp is 10 m long and makes an angle of 30 degrees with the horizontal ground. Calculate the exact height of the top of the ramp above the ground.
Diagram NOT accurately drawn
(Total for Question 14 is 2 marks)
15
A flagpole casts a shadow 12 m long on horizontal ground. The angle of elevation of the sun is 60 degrees. Calculate the exact height of the flagpole. Give your answer in the form k √3.
Diagram NOT accurately drawn
(Total for Question 15 is 3 marks)
16
Triangle XYZ has XY = 8 cm, XZ = 10 cm and angle YXZ = 30 degrees. Calculate the exact area of triangle XYZ.
Diagram NOT accurately drawn
(Total for Question 16 is 3 marks)
17
Triangle LMN has LM = 6 cm, LN = 6 cm and angle MLN = 60 degrees. Calculate the exact area of triangle LMN. Give your answer in the form k √3.
Diagram NOT accurately drawn
(Total for Question 17 is 3 marks)
18
Triangle ABC is equilateral with each side of length 10 cm. M is the midpoint of BC.
Diagram NOT accurately drawn
(a)Calculate the exact length of AM, the height of the triangle. Give your answer in the form k √3.(3)
(b)Hence calculate the exact area of triangle ABC. Give your answer in the form k √3.(2)
(Total for Question 18 is 5 marks)
19
Show that 1/cos(30) can be written in the form (2 √3)/3
(Total for Question 19 is 3 marks)
20
Show that (sin 45 + cos 45)2 = 2
(Total for Question 20 is 3 marks)
21
A right-angled triangle has a hypotenuse of length 4 √3 cm and one other angle of 30 degrees. Work out the exact perimeter of the triangle. Give your answer in the form a + b √3, where a and b are integers.
Diagram NOT accurately drawn
(Total for Question 21 is 5 marks)
Mark scheme · 5.20 Exact trig values
Question 1
(a) B1√3/2 oe
(a) Answer: √3/2
(b) B1√3/3 oe, accept 1/√3
(b) Answer: √3/3
(c) B1√2/2 oe
(c) Answer: √2/2
(d) B1√2/2 oe
(d) Answer: √2/2
(e) B1 1 cao
(e) Answer: 1
(f) B1√3/2 oe
(f) Answer: √3/2
(g) B1 1/2 oe
(g) Answer: 1/2
(h) B1√3 cao
(h) Answer: √3
Question 2
(a) B1 0 cao
(a) Answer: 0
(b) B1 1 cao
(b) Answer: 1
(c) B1 1 cao
(c) Answer: 1
(d) B1 0 cao
(d) Answer: 0
Question 3
B1 A
Answer: A (1/2)
Question 4
M1 sin60 = √3/2 and cos60 = 1/2 substituted
A1 (√3 + 1)/2 oe cao
Answer: (√3 + 1)/2
Question 5
M1 correct substitution 4(1/2) - 2(1/2)
A1 1 cao
Answer: 1
Question 6
M1 tan60 = √3 and tan30 = √3/3 (or 1/√3) stated or used
M1√3 x √3/3 = 3/3 simplification seen
A1 1 cao
Answer: 1
Question 7
M1 recognise √3/2 as an exact sine value
A1 x = 60 cao
Answer: x = 60
Question 8
M1 recognise √2/2 as an exact cosine value
A1 x = 45 cao
Answer: x = 45
Question 9
M1 recognise √3/3 (or 1/√3) as an exact tangent value
A1 x = 30 cao, with reason tan30 = √3/3
Answer: x = 30
Question 10
M1 BC = AC x sin(30) or equivalent correct trig equation
A1 BC = 4 cm cao
Answer: BC = 4 cm
Question 11
M1 tan(60) = QR/PQ or equivalent correct trig equation set up
M1 QR = 5 x tan(60) with tan60 = √3 substituted
A1 QR = 5 √3 cm cao
Answer: QR = 5 √3 cm
Question 12
M1 sin(45) = EF/DF or equivalent correct trig equation set up
M1 DF = 6 / (√2/2) rationalised to 12/√2 or equivalent
A1 DF = 6 √2 cm cao
Answer: DF = 6 √2 cm
Question 13
M1 hypotenuse = 7 / sin(45) or √72 + 72 set up
A1 7 √2 cm cao
Answer: 7 √2 cm
Question 14
M1 height = 10 x sin(30)
A1 height = 5 m cao
Answer: 5 m
Question 15
M1 tan(60) = height/12 or equivalent correct trig equation set up
M1 height = 12 x tan(60) with tan60 = √3 substituted
A1 height = 12 √3 m cao
Answer: 12 √3 m
Question 16
M1 Area = 1/2 x 8 x 10 x sin(30) correct formula with values substituted
M1 sin(30) = 1/2 used correctly
A1 Area = 20 cm2 cao
Answer: 20 cm2
Question 17
M1 Area = 1/2 x 6 x 6 x sin(60) correct formula with values substituted
M1 sin(60) = √3/2 used correctly
A1 Area = 9 √3 cm2 cao
Answer: 9 √3 cm2
Question 18
(a) M1 tan(60) = AM/5 or equivalent correct trig equation using right triangle ABM
(a) M1 AM = 5 x tan(60) with tan60 = √3 substituted
(a) A1 AM = 5 √3 cm cao
(a) Answer: 5 √3 cm
(b) M1 Area = 1/2 x 10 x 5 √3 ft their AM from part (a)