Standard +0.3 This is a standard trigonometric equation requiring the technique of dividing by cos to convert tan to sin/cos, then solving the resulting equation. It involves multiple steps (rewriting tan, rearranging to quadratic in sin, solving, finding angles in range) but uses routine A-level methods without requiring novel insight. The angle shift of 43° adds minor complexity but is standard bookwork. Slightly above average due to the multi-step nature and need to carefully handle the range, but well within typical P2 expectations.
In this question you must show all stages of your working.
Solutions relying entirely on calculator technology are not acceptable.
Solve, for \(-180° < \theta \leq 180°\), the equation
$$3\tan(\theta + 43°) = 2\cos(\theta + 43°)$$
[6]
3 t a n ( 4 3 ) 2 c o s ( 4 3 ) 3 2 c o s ( 4 3 ) + = + = +
c o s 4 3 +
Answer
Marks
3 s i n ( 4 3 ) 2 c o s 2 ( 4 3 ) + = +
M1
3 s i n ( 4 3 ) 2 ( 1 s i n 2 ( 4 3 ) ) + = − +
M1
2 s i n 2 ( 4 3 ) 3 s i n ( 4 3 ) 2 0 + + + − =
Answer
Marks
( 2 s i n ( 4 3 ) 1 ) ( s i n ( 4 3 ) 2 ) 0 s i n ( 4 3 ) . .. + − + + = + =
M1
1
sin(+43)=
Answer
Marks
2
A1
" 1 "
a r c s i n 4 3 = −
Answer
Marks
2
M1
1 3 , 1 0 7 = −
A1
(6)
(6 marks)
Notes:
sin..
M1: Uses tan..= and multiplies through to form an equation of the form A s i n ... = B c o s 2 ...
cos..
Condone poor notation e.g.:
sin
3tan(+43)=2cos(+43)3 (+43)=2cos(+43)
cos
3 s i n ( 4 3 ) 2 c o s 2 ( 4 3 ) + = + (with or without brackets)
M1: Applies Pythagorean identity to obtain a 3 term quadratic equation in sin.
Allow use of cos2...=1sin2...
M1: Solves a 3 term quadratic in sin(+43) by any valid means.
This may be implied by at least one correct root for their quadratic.
Allow if they have sin (+43)= x or another variable or e.g. sin α where 4 3 = +
A1: Correct value of s i n ( 4 3 ) + . If s i n ( 4 3 ) x + = is used, it must be clear they mean
s i n ( 4 3 ) + but this may be implied if they have e.g. sin α = ½ where 4 3 = +
1
If x = is left as a final answer it is A0.
2
Answer
Marks
Guidance
M1: Correct method for solving s i n ( 4 3 ) k ,
k
1 + = , look for use of inverse sine followed by
−1(their
subtraction of 43 from sin k). Implied by one correct solution for their k
Do not allow mixing of degrees and radians for this mark.
A1: Correct solutions and no others in the range.
Answer
Marks
Guidance
Question
Scheme
Marks
Question 5:
5 | s i n (( 4 3 )) +
3 t a n ( 4 3 ) 2 c o s ( 4 3 ) 3 2 c o s ( 4 3 ) + = + = +
c o s 4 3 +
3 s i n ( 4 3 ) 2 c o s 2 ( 4 3 ) + = + | M1
3 s i n ( 4 3 ) 2 ( 1 s i n 2 ( 4 3 ) ) + = − + | M1
2 s i n 2 ( 4 3 ) 3 s i n ( 4 3 ) 2 0 + + + − =
( 2 s i n ( 4 3 ) 1 ) ( s i n ( 4 3 ) 2 ) 0 s i n ( 4 3 ) . .. + − + + = + = | M1
1
sin(+43)=
2 | A1
" 1 "
a r c s i n 4 3 = −
2 | M1
1 3 , 1 0 7 = − | A1
(6)
(6 marks)
Notes:
sin..
M1: Uses tan..= and multiplies through to form an equation of the form A s i n ... = B c o s 2 ...
cos..
Condone poor notation e.g.:
sin
3tan(+43)=2cos(+43)3 (+43)=2cos(+43)
cos
3 s i n ( 4 3 ) 2 c o s 2 ( 4 3 ) + = + (with or without brackets)
M1: Applies Pythagorean identity to obtain a 3 term quadratic equation in sin.
Allow use of cos2...=1sin2...
M1: Solves a 3 term quadratic in sin(+43) by any valid means.
This may be implied by at least one correct root for their quadratic.
Allow if they have sin (+43)= x or another variable or e.g. sin α where 4 3 = +
A1: Correct value of s i n ( 4 3 ) + . If s i n ( 4 3 ) x + = is used, it must be clear they mean
s i n ( 4 3 ) + but this may be implied if they have e.g. sin α = ½ where 4 3 = +
1
If x = is left as a final answer it is A0.
2
M1: Correct method for solving s i n ( 4 3 ) k , | k | 1 + = , look for use of inverse sine followed by
−1(their
subtraction of 43 from sin k). Implied by one correct solution for their k
Do not allow mixing of degrees and radians for this mark.
A1: Correct solutions and no others in the range.
Question | Scheme | Marks
In this question you must show all stages of your working.
Solutions relying entirely on calculator technology are not acceptable.
Solve, for $-180° < \theta \leq 180°$, the equation
$$3\tan(\theta + 43°) = 2\cos(\theta + 43°)$$
[6]
\hfill \mbox{\textit{Edexcel P2 2022 Q5 [6]}}