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GCSE · Trigonometry

5.20D Exact trig values: Fluency and Exam Drill

EDEXCEL 1MA1 · Calculators not allowed · about 55 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

Write down the exact value of sin 0 degrees.

(Total for Question 1 is 1 mark)

Write down the exact value of cos 0 degrees.

(Total for Question 2 is 1 mark)

Write down the exact value of tan 0 degrees.

(Total for Question 3 is 1 mark)

Write down the exact value of sin 90 degrees.

(Total for Question 4 is 1 mark)

Write down the exact value of cos 90 degrees.

(Total for Question 5 is 1 mark)

Write down the exact value of tan 45 degrees.

(Total for Question 6 is 1 mark)

Write down the exact value of sin 45 degrees, giving your answer as a surd in its simplest form.

(Total for Question 7 is 1 mark)

Write down the exact value of cos 45 degrees, giving your answer as a surd in its simplest form.

(Total for Question 8 is 1 mark)

Write down the exact value of sin 30 degrees.

(Total for Question 9 is 1 mark)

Write down the exact value of cos 60 degrees.

(Total for Question 10 is 1 mark)

Write down the exact value of sin 60 degrees, giving your answer as a surd in its simplest form.

(Total for Question 11 is 1 mark)

Write down the exact value of cos 30 degrees, giving your answer as a surd in its simplest form.

(Total for Question 12 is 1 mark)

Write down the exact value of tan 60 degrees, giving your answer as a surd in its simplest form.

(Total for Question 13 is 1 mark)

Write down the exact value of tan 30 degrees, giving your answer in the form √a/b, where a and b are positive integers.

(Total for Question 14 is 1 mark)

Explain why tan 90 degrees is undefined.

(Total for Question 15 is 1 mark)

Work out the exact value of sin 30 degrees + cos 60 degrees. Show your working.

(Total for Question 16 is 2 marks)

Work out the exact value of (sin 45 degrees)² + (cos 45 degrees)^2. Give your answer as an integer.

(Total for Question 17 is 2 marks)

Given that tan(θ) = √3 for 0 ≤ θ ≤ 90, find the exact value of θ. Hence work out the exact value of sin(θ) + cos(θ), giving your answer as a single fraction in its simplest form.

(Total for Question 18 is 3 marks)

A point P lies on a circle with centre O at the origin and radius 8 cm. The line segment OP makes an angle of 60 degrees with the positive x-axis. Work out the exact coordinates of P, giving each coordinate in its simplest exact form.

(Total for Question 19 is 4 marks)

Show that sin 60 degrees x cos 30 degrees + cos 60 degrees x sin 30 degrees = 1. You must show each step of your working.

(Total for Question 20 is 4 marks)

A student claims that sin 45 degrees + sin 45 degrees is equal to sin 90 degrees. By finding the exact value of each side, determine whether the student is correct. You must show your working and give a reason for your conclusion.

(Total for Question 21 is 3 marks)

A rectangle has a diagonal of exact length 8 cm. The diagonal makes an angle of 30 degrees with one of the longer sides of the rectangle. Work out the exact area of the rectangle. Give your answer in the form a*√b cm², where a and b are integers and b is as small as possible.

(Total for Question 22 is 4 marks)

Solve the equation 2cos(θ) = √3 for 0 ≤ θ ≤ 90, giving an exact value of θ. Using your value of θ, work out the exact value of 4(sin(θ))² - 1.

(Total for Question 23 is 5 marks)

Show that 1/(tan 30 degrees) can be written as √3. Hence work out the exact value of (1/(tan 30 degrees))² + (tan 60 degrees)², giving your answer as an integer.

(Total for Question 24 is 5 marks)

Mark scheme · 5.20D Exact trig values: Fluency and Exam Drill

Question 1

B1 0 cao
Answer: 0

Question 2

B1 1 cao
Answer: 1

Question 3

B1 0 cao
Answer: 0

Question 4

B1 1 cao
Answer: 1

Question 5

B1 0 cao
Answer: 0

Question 6

B1 1 cao
Answer: 1

Question 7

B1 √2/2 oe, accept 1/√2
Answer: √2/2

Question 8

B1 √2/2 oe, accept 1/√2
Answer: √2/2

Question 9

B1 1/2 oe cao
Answer: 1/2

Question 10

B1 1/2 oe cao
Answer: 1/2

Question 11

B1 √3/2 oe
Answer: √3/2

Question 12

B1 √3/2 oe
Answer: √3/2

Question 13

B1 √3 cao
Answer: √3

Question 14

B1 √3/3 cao (a = 3, b = 3); rationalised form required for the mark
Answer: √3/3

Question 15

B1 correct reference to tan θ = sin θ / cos θ and cos90 = 0, so the calculation would involve dividing by zero (oe: adjacent side has length 0)
Answer: tan90 is undefined because tan θ = sin θ / cos θ, and cos90 = 0, so the calculation would require dividing by zero.

Question 16

M1 sin30 = 1/2 and cos60 = 1/2 both seen or used
A1 1 cao
Answer: 1

Question 17

M1 substitutes sin45 = √2/2 and cos45 = √2/2 and squares each term to reach 1/2 + 1/2 (or equivalent, e.g. 2/4 + 2/4)
A1 1 cao
Answer: 1

Question 18

M1 identifies θ = 60 (from the exact value tan60 = √3)
M1 substitutes sin60 = √3/2 and cos60 = 1/2
A1 (√3 + 1)/2 oe cao
Answer: θ = 60 degrees; sin60 + cos60 = (√3 + 1)/2

Question 19

M1 sets up x-coordinate = 8 x cos60
A1 x = 4 cao
M1 sets up y-coordinate = 8 x sin60, using sin60 = √3/2
A1 y = 4sqrt(3) cao
Answer: P = (4, 4sqrt(3))

Question 20

M1 substitutes all four exact values: sin60 = √3/2, cos30 = √3/2, cos60 = 1/2, sin30 = 1/2
M1 correctly multiplies √3/2 x √3/2 = 3/4
M1 correctly multiplies 1/2 x 1/2 = 1/4
A1 3/4 + 1/4 = 1 cso, answer given
Answer: 1 (shown)

Question 21

M1 finds sin45 + sin45 = 2 x √2/2 = √2
M1 finds sin90 = 1
A1 correct conclusion with reason, e.g. 'not correct because √2 does not equal 1' (dependent on both previous marks)
Answer: Not correct: sin45 + sin45 = √2, but sin90 = 1, and √2 does not equal 1.

Question 22

M1 sets up longer side = 8 x cos30
A1 longer side = 4sqrt(3) cm cao
M1 sets up shorter side = 8 x sin30, giving shorter side = 4 cm
A1 area = 16sqrt(3) cm² cao (a = 16, b = 3)
Answer: 16sqrt(3) cm² (a = 16, b = 3)

Question 23

M1 rearranges to cos(θ) = √3/2
A1 θ = 30 (degrees) cao
M1 substitutes sin30 = 1/2 into the expression
M1 squares and multiplies to reach 4 x 1/4 = 1
A1 0 cao
Answer: θ = 30 degrees; 4(sin(θ))² - 1 = 0

Question 24

M1 writes 1/(tan30) = 1/(√3/3), or equivalently cos30/sin30 = (√3/2)/(1/2)
A1 simplifies to √3 cso, answer given
M1 squares 1/(tan30) to get (√3)² = 3
M1 squares tan60 to get (√3)² = 3
A1 6 cao
Answer: 1/(tan30) = √3 (shown); (1/(tan30))² + (tan60)² = 6

Exact trig values: Fluency and Exam Drill - Worksheets, Questions and Revision