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5.20H Exact trig values: Higher Tier Practice

EDEXCEL 1MA1 · Calculator allowed · about 75 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

Write down the exact values of each of the following.

a) sin 30 degrees
b) cos 45 degrees
c) tan 60 degrees

(a)Write down sin 30 degrees.(1)

(b)Write down cos 45 degrees.(1)

(c)Write down tan 60 degrees.(1)

(Total for Question 1 is 3 marks)

Show that sin 15 degrees = (√6 - √2) / 4. You may use exact values for sin 30, cos 30, sin 45 and cos 45.

(main)Show that sin 15 degrees = (√6 - √2) / 4.(3)

(Total for Question 2 is 3 marks)

Show that tan 15 degrees = 2 - √3. You may use exact values or angle formulae.

(main)Show that tan 15 degrees = 2 - √3.(3)

(Total for Question 3 is 3 marks)

Simplify the following expression to an exact surd form, showing working.

(1 - cos 60 degrees) / sin 30 degrees

(main)Simplify (1 - cos 60) / sin 30 to an exact value.(3)

(Total for Question 4 is 3 marks)

Solve for θ, giving exact answers in degrees, in the interval 0 ≤ θ < 180 degrees.

sin θ = √3/2

(main)Solve sin θ = √3/2 for 0 ≤ θ < 180 degrees.(2)

(Total for Question 5 is 2 marks)

Write down the exact value of cos 75 degrees. Give a brief reason.

(main)Write down cos 75 degrees and give a reason.(2)

(Total for Question 6 is 2 marks)

Show that tan 75 degrees = 2 + √3. You may use exact values or angle formulae.

(main)Show that tan 75 degrees = 2 + √3.(3)

(Total for Question 7 is 3 marks)

In triangle ABC, angle A = 45 degrees, angle B = 75 degrees and angle C = 60 degrees. The side AB = 1 (the side opposite angle C). Use the sine rule to find the exact lengths of BC and AC, giving answers in simplified surd form.

(main)Use the sine rule to find BC and AC, exact surd form, given AB = 1 and angles A = 45, B = 75, C = 60.(4)

(Total for Question 8 is 4 marks)

Show that sin 18 degrees = (√5 - 1) / 4.

You may use algebraic methods or known exact cosine values; show all steps clearly.

(main)Derive sin 18 degrees = (√5 - 1) / 4, showing working.(8)

(Total for Question 9 is 8 marks)

Solve the equation tan 2theta = √3 for 0 ≤ θ < 360 degrees. Give all solutions in degrees and show your working clearly.

(main)Solve tan 2theta = √3 for 0 ≤ θ < 360 degrees, giving all solutions.(9)

(Total for Question 10 is 9 marks)

Mark scheme · 5.20H Exact trig values: Higher Tier Practice

Question 1

(a) B1 1/2 cao
(a) Answer: 1/2
(b) B1 √2/2 cao
(b) Answer: √2/2
(c) B1 √3 cao
(c) Answer: √3

Question 2

(main) M1 use sine difference formula sin(45 - 30) = sin45 cos30 - cos45 sin30 or equivalent
(main) M1 substitute exact values: (√2/2)(√3/2) - (√2/2)(1/2) leading to (√6 - √2)/4
(main) A1 (√6 - √2)/4 cao, cso
(main) Answer: (√6 - √2) / 4

Question 3

(main) M1 use tan(45 - 30) = (tan45 - tan30) / (1 + tan45 tan30) or equivalent
(main) M1 substitute tan45 = 1 and tan30 = 1/√3 and simplify to (sqrt3 -1)/(sqrt3 +1) or equivalent
(main) A1 2 - √3 cao
(main) Answer: 2 - √3

Question 4

(main) M1 substitute cos60 = 1/2 and sin30 = 1/2 or show equivalent method
(main) M1 compute numerator 1 - 1/2 = 1/2 and divide by 1/2
(main) A1 1 cao
(main) Answer: 1

Question 5

(main) M1 identify principal angle 60 degrees and use symmetry to find second solution 120 degrees
(main) A1 θ = 60 degrees and 120 degrees cao
(main) Answer: θ = 60 degrees or 120 degrees

Question 6

(main) B1 cos75 = cos(45 + 30) or note cos75 = sin15 oe
(main) A1 cos75 = (√6 - √2) / 4 cao
(main) Answer: cos 75 degrees = (√6 - √2) / 4

Question 7

(main) M1 use tan(45 + 30) = (tan45 + tan30) / (1 - tan45 tan30) or equivalent
(main) M1 substitute tan45 = 1 and tan30 = 1/√3 and simplify to (sqrt3 +1)/(sqrt3 -1) or equivalent
(main) A1 2 + √3 cao
(main) Answer: 2 + √3

Question 8

(main) M1 use sine rule: a/sinA = b/sinB = c/sinC and substitute c = 1, sin60 = √3/2
(main) M1 find BC = a = sin45 / sin60 and simplify to sqrt6/3 or equivalent
(main) M1 find AC = b = sin75 / sin60 and simplify to an exact surd
(main) A1 BC = sqrt6/3 and AC = (3sqrt2 + sqrt6)/6 cao
(main) Answer: BC = sqrt6/3, AC = (3sqrt2 + sqrt6)/6

Question 9

(main) M1 use a suitable exact relation such as cos36 = (sqrt5 +1)/4 or derive a polynomial for cos36 using double-angle or pentagon symmetry
(main) M1 use relationship sin18 = √(1 - cos36)/2 or equivalent identity
(main) M1 substitute cos36 = (sqrt5 +1)/4 and simplify under the square root
(main) M1 simplify to (√5 - 1)/4 as positive root (since sin18 is positive)
(main) M1 give algebraic justification for selecting the positive root (angle in first quadrant)
(main) A1 sin18 = (√5 - 1)/4 cao, cso
(main) M1 alternative valid route: start from 5-fold angle equation for sine/cosine and factor to get the same value
(main) A1 final exact form shown clearly
(main) Answer: sin 18 degrees = (√5 - 1) / 4

Question 10

(main) M1 recognise general solution for tan x = √3 is x = 60 + k*180 degrees
(main) M1 set 2theta = 60 + k*180 and solve for θ giving θ = 30 + k*90 degrees
(main) M1 find integer k values that give θ in 0 ≤ θ < 360 and list candidates
(main) M1 substitute k = 0,1,2,3 to produce θ = 30, 120, 210, 300 degrees
(main) M1 check that each value is distinct and within the interval
(main) A1 state final set θ = 30, 120, 210, 300 degrees cao
(main) M1 optional check: show tan 2theta equals sqrt3 for each listed θ (e.g. 2*30=60, tan60 = sqrt3) ft
(main) M1 note that no other k produce values in the interval
(main) A1 complete solution set given
(main) Answer: θ = 30 degrees, 120 degrees, 210 degrees, 300 degrees

Exact trig values: Higher Tier Practice - Worksheets, Questions and Revision