Further Trigonometric Identities and Equations: Fluency and Exam Drill - Worksheets, Questions and Revision

19 original exam-style questions - 4 pages of questions with a full mark scheme - free printable PDF.

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GCSE · Trigonometry

G3D Further Trigonometric Identities and Equations: Fluency and Exam Drill

AQA 8365 · Calculator allowed · about 50 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

Write down the exact value of cos(45 degrees).

(Total for Question 1 is 1 mark)

Write down the exact value of tan(60 degrees).

(Total for Question 2 is 1 mark)

Angle x is acute, and sin(x) = 5/13. Find the exact value of cos(x).

(Total for Question 3 is 2 marks)

Simplify sin²(x) + cos²(x) - 3.

(Total for Question 4 is 1 mark)

State the value of x, with 0 ≤ x ≤ 180, for which sin(x) = 1.

(Total for Question 5 is 1 mark)

Which of the following is equivalent to 1 - sin²(x)?

A) cos²(x)
B) tan²(x)
C) -cos²(x)
D) sin²(x) - 1

(Total for Question 6 is 1 mark)

Solve tan(x) = 1 for 0 ≤ x ≤ 180.

(Total for Question 7 is 1 mark)

Solve sin(x) = 0.5 for 0 ≤ x ≤ 360, giving all solutions.

(Total for Question 8 is 2 marks)

Solve cos(x) = -0.5 for 0 ≤ x ≤ 360, giving all solutions.

(Total for Question 9 is 2 marks)

Given that cos(x) = 4/5 and x is acute, find the exact value of tan(x).

(Total for Question 10 is 2 marks)

Show that (1 - sin²(x))/cos(x) = cos(x), for cos(x) not equal to 0.

(Total for Question 11 is 2 marks)

State the value of sin²(30 degrees) + cos²(30 degrees).

(Total for Question 12 is 1 mark)

Solve 2cos(x) = 1 for 0 ≤ x ≤ 360, giving all solutions.

(Total for Question 13 is 2 marks)

Solve 5cos(x) = 2 for 0 ≤ x ≤ 360, giving your answers correct to 1 decimal place.

(Total for Question 14 is 3 marks)

Given that sin(θ) = 0.6, where θ is obtuse (90 < θ < 180), use sin²(θ) + cos²(θ) = 1 to find the value of cos(θ).

(Total for Question 15 is 3 marks)

Prove that (1 + sin(x))(1 - sin(x)) = cos²(x) for all values of x.

(Total for Question 16 is 3 marks)

A ramp makes an angle x with the horizontal, where 0 < x < 90, and satisfies 3sin(x) = 2cos(x). Find the exact value of tan(x), and hence find x, giving your answer correct to 1 decimal place.

(Total for Question 17 is 3 marks)

Solve 2cos²(x) + 3cos(x) - 2 = 0 for 0 ≤ x ≤ 360, giving all solutions.

(Total for Question 18 is 4 marks)

A force vector makes an angle x with a reference axis, where sin(x) + cos(x) = 7/5.

(a)Show that 2sin(x)cos(x) = 24/25.(3)

(b)Hence find the value of (sin(x) - cos(x))².(2)

(Total for Question 19 is 5 marks)

Mark scheme · G3D Further Trigonometric Identities and Equations: Fluency and Exam Drill

Question 1

B1 √2/2 oe (e.g. 1/√2)
Answer: √2/2

Question 2

B1 √3 oe
Answer: √3

Question 3

M1 use sin²(x) + cos²(x) = 1 to get cos²(x) = 1 - (5/13)² = 144/169
A1 cos(x) = 12/13 (positive root, since x is acute), cso
Answer: 12/13

Question 4

B1 -2 cao
Answer: -2

Question 5

B1 x = 90 cao
Answer: x = 90

Question 6

B1 A) cos²(x)
Answer: A) cos²(x)

Question 7

B1 x = 45 cao (the next solution, 225, lies outside the given range)
Answer: x = 45

Question 8

M1 arcsin(0.5) = 30 (principal value)
A1 x = 30 and x = 150 (using 180 - 30), both required cao
Answer: x = 30, 150

Question 9

M1 reference angle arccos(0.5) = 60, and recognises cosine is negative in the second and third quadrants
A1 x = 120 and x = 240, both required cao
Answer: x = 120, 240

Question 10

M1 use sin²(x) + cos²(x) = 1 to find sin(x) = 3/5 (positive root, since x is acute)
A1 tan(x) = 3/4 oe cao
Answer: 3/4

Question 11

M1 use sin²(x) + cos²(x) = 1 to write 1 - sin²(x) = cos²(x)
A1 cos²(x)/cos(x) = cos(x), fully shown, cso
Answer: cos(x) (shown)

Question 12

B1 1 cao
Answer: 1

Question 13

M1 rearrange to cos(x) = 0.5
A1 x = 60 and x = 300, both required cao
Answer: x = 60, 300

Question 14

M1 rearrange to cos(x) = 0.4
A1 x = 66.4 (awrt, principal value)
A1 x = 293.6 (ft: 360 - previous value, awrt)
Answer: x = 66.4 or x = 293.6

Question 15

M1 correctly rearranges sin²(θ) + cos²(θ) = 1 to cos²(θ) = 1 - sin²(θ)
M1 cos²(θ) = 1 - 0.6² = 1 - 0.36 = 0.64
A1 cos(θ) = -0.8 (negative root selected since θ is obtuse) cao
Answer: cos(θ) = -0.8

Question 16

M1 expand the LHS using difference of two squares: 1 - sin²(x)
M1 use sin²(x) + cos²(x) = 1 to write 1 - sin²(x) = cos²(x)
A1 fully shown: = cos²(x), cso
Answer: cos²(x) (shown)

Question 17

M1 divide both sides by cos(x) to form 3tan(x) = 2, hence tan(x) = 2/3
M1 arctan(2/3) = 33.7 (awrt, principal value)
A1 x = 33.7 cao (awrt 1 d.p.)
Answer: tan(x) = 2/3, x = 33.7 degrees

Question 18

M1 let c = cos(x) and form 2c² + 3c - 2 = 0
M1 factorise (2c - 1)(c + 2) = 0, giving c = 0.5 or c = -2
A1 reject c = -2, since -1 ≤ cos(x) ≤ 1 for all x
A1 x = 60 and x = 300 (both required, from c = 0.5) cao
Answer: x = 60 or x = 300 (cos(x) = -2 is rejected, since cos(x) can never be less than -1)

Question 19

(a) M1 square both sides: (sin(x) + cos(x))² = (7/5)²
(a) M1 expand LHS and use sin²(x) + cos²(x) = 1 to write 1 + 2sin(x)cos(x) = 49/25
(a) A1 2sin(x)cos(x) = 49/25 - 1 = 24/25, cso
(a) Answer: 2sin(x)cos(x) = 24/25 (shown)
(b) M1 use (sin(x) - cos(x))² = 1 - 2sin(x)cos(x), ft from part (a)
(b) A1 = 1/25 oe (0.04) cao
(b) Answer: 1/25

Question 19

5 marks

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Further Trigonometric Identities and Equations: Fluency and Exam Drill - Worksheets, Questions and Revision

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G3D Further Trigonometric Identities and Equations: Fluency and Exam Drill

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Question 19

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