MATH IN FOCUS WITH RAJA NADEEM AMIN 03215862394
O & A Level campus
Trigonometric Functions
Q1
(N 2004/P1)
The function 𝑓(𝑥) = 5sin2 𝑥 + 3 cos2 𝑥 is defined for the domain 0 ≤ 𝑥 ≤ 𝜋.
(i)
Express 𝑓(𝑥) in the form 𝑎 + 𝑏 sin2 𝑥, stating the values of ‘a’ and ‘b’.
(ii)
Hence find the values of x for which 𝑓(𝑥) = 7𝑠𝑖𝑛𝑥.
(iii)
State the range of 𝑓(𝑥).
Q2
(J2005/P1/Q7)
A function f is defined by (𝑥) = 3 − 2𝑠𝑖𝑛𝑥 , for 00 ≤ x ≤3600 .
(i)
Find the range of 𝑓.
(ii)
Sketch the graph of 𝑦 = 𝑓(𝑥)
A function g is defined by 𝑔(𝑥) = 3 – 2𝑠𝑖𝑛𝑥, for 00 ≤ x ≤A0 , where A is a constant.
(iii)
State the largest value of A for which g has an inverse.
(iv)
When A has this value, obtain an expression, in terms of x, for g-1(x).
Q3.
(J2006/P1/Q2)
[1]
[2]
𝑠𝑖𝑛 2𝑥 + 3 𝑐𝑜𝑠 2𝑥 = 0,
Solve the equation
for 00 ≤ 𝑥 ≤ 1800 .
Q4.
[2]
[2]
[4]
(J2007/P1/Q8)
The function f is defined by 𝑓(𝑥) = 𝑎 + 𝑏𝑐𝑜𝑠2𝑥, for 00 ≤ x ≤ π. It is given that
𝜋
𝑓(0) = −1 and 𝑓 ( 2 ) = 7
(i)
(ii)
(iii)
Q5.
Find the values of a and b.
Find the x coordinates of the points where the curve 𝑦 = 𝑓(𝑥) intersects the x-axis.
Sketch the graph of 𝑦 = 𝑓(𝑥).
(J2007/P1/Q3)
1−tan2 𝑥
Prove the identity 1+tan2 𝑥 ≡ 1 − 2 sin2 𝑥.
[3]
[3]
[2]
[4]
MATH IN FOCUS WITH RAJA NADEEM AMIN 03215862394
Q6.(j2008/P1/Q2)
(i) Show that the equation 2 tan2 𝜃 𝑐𝑜𝑠 𝜃 = 3 can be written in the form
Q7.
2 cos 2 𝜃 + 3 𝑐𝑜𝑠 𝜃 − 2 = 0.
[2]
(ii) Hence solve the equation 2 tan2 𝜃 𝑐𝑜𝑠 𝜃 = 3, for 00 ≤ 𝜃 ≤ 3600 .
[3]
(J2009/P1/Q4)
The diagram shows the graph of
𝑦 = 𝑎 𝑠𝑖𝑛(𝑏𝑥) + 𝑐 for 0 ≤ 𝑥 ≤ 2𝜋.
(i) Find the values of a, b and c.
(ii) Find the smallest value of x in the interval 0 ≤ 𝑥 ≤ 2𝜋for which 𝑦 = 0.
Q8.
[3]
[3]
(J2009/P1/Q1)
𝑠𝑖𝑛𝑥
𝑠𝑖𝑛𝑥
Prove the identity 1−𝑠𝑖𝑛𝑥 − 1+𝑠𝑖𝑛𝑥 ≡ 2 tan2 𝑥
Q9.
[3]
(J2010/P1/Q1)
The acute angle x radians is such that tan x = k, where k is a positive constant. Express, in terms of k,
(i) 𝑡𝑎𝑛(𝜋 − 𝑥),
[1]
𝜋
Q10.
(ii) 𝑡𝑎𝑛 ( 2 – 𝑥),
[1]
(iii) sin x.
[2]
(J2010/P1/Q5)
The function 𝑓 is such that 𝑓(𝑥) = 2 sin2 𝑥 − 3 𝑐𝑜𝑠2𝑥 for 0 ≤ 𝑥 ≤ 𝜋.
(i)
(ii)
(iii)
Express 𝑓(𝑥) in the form 𝑎 + 𝑏 cos 2 𝑥, stating the values of a and b.
State the greatest and least values of 𝑓(𝑥).
Solve the equation (𝑥) + 1 = 0 .
[2]
[2]
[3]
MATH IN FOCUS WITH RAJA NADEEM AMIN 03215862394
Q11.
(J2011/P1/Q5)
𝑐𝑜𝑠𝜃
(i)
Prove the identity 𝑡𝑎𝑛𝜃(
(ii)
Hence solve the equation 𝑡𝑎𝑛𝜃(
Q12.
1
≡ 1 + 𝑠𝑖𝑛𝜃
1−𝑠𝑖𝑛𝜃 )
𝑐𝑜𝑠𝜃
1−𝑠𝑖𝑛𝜃 )
[3]
= 4 for 00 ≤ 𝜃 ≤ 3600
(J2012/P11/Q1)
Solve the equation 𝑠𝑖𝑛 2𝑥 = 2 𝑐𝑜𝑠 2𝑥, for 00 ≤ 𝑥 ≤ 1800 .
Q13.
[3]
[4]
(J2012/P11/Q7)
(a) The first two terms of an arithmetic progression are 1 and cos 2 𝑥 respectively. Show that the sum of
the first ten terms can be expressed in the form 𝑎 – 𝑏𝑠𝑖𝑛2 𝑥, where a and b are constants to be found.
[3]
1
(b) The first two terms of a geometric progression are 1 and 3 tan2 𝜃 respectively, where
𝜋
0 < 𝜃 < 2.
Q14.
(i)
Find the set of values of 𝜃 for which the progression is convergent.
(ii)
Find the exact value of the sum to infinity when 𝜃 = 6 .
𝜋
[2]
[2]
(J2012/P12/Q5)
(i) Prove the identity
[2]
1
1
𝑡𝑎𝑛𝑥 +
≡
𝑡𝑎𝑛𝑥 𝑠𝑖𝑛𝑥𝑐𝑜𝑠𝑥
(ii)
Solve the equation
for 0 ≤ 𝑥 ≤ 1800 .
Q15.
(i)
(ii)
(iii)
1
= 1 + 3𝑡𝑎𝑛𝑥
𝑠𝑖𝑛𝑥𝑐𝑜𝑠𝑥
[4]
The function f is such that 𝑓(𝑥) = 𝑎 – 𝑏 𝑐𝑜𝑠𝑥 for 00 ≤ x ≤3600, where a and b are positive constants.
The maximum value of 𝑓(𝑥) is 10 and the minimum value is -2.
Find the value of a and b.
Solve the equation 𝑓(𝑥) = 0.
Sketch the graph of 𝑦 = 𝑓(𝑥).
[3]
[3]
[2]
MATH IN FOCUS WITH RAJA NADEEM AMIN 03215862394
Q16.
(J2014/P11/Q1)
The diagram shows part of the graph of y = a + bsin x. State the values of the constants a and b. [2]
Q17.
(J2015/P11/8)
𝑥
The function 𝑓: 𝑥 → 5 + 3 cos (2) is defined for 0 ≤ 𝑥 ≤ 2𝜋
(i)
(ii)
(iii)
(iv)
Q18.
Solve the equation𝑓(𝑥) = 7, giving your answer correct to 2 decimal places.
Sketch the graph of 𝑦 = 𝑓(𝑥).
Explain why f has an inverse.
Obtain an expression for 𝑓 −1 (𝑥).
[3]
[2]
[1]
[3]
(J2016/P11/Q11)
𝜋
𝜋
The function f is defined by 𝑓 ∶ 𝑥 → 4 𝑠𝑖𝑛 𝑥 − 1 for – 2 ≤ 𝑥 ≤ 2 .
(i)
(ii)
(iii)
(iv)
Q19.
State the range of f.
[2]
Find the coordinates of the points at which the curve 𝑦 = 𝑓(𝑥) intersects the coordinate axes.[3]
Sketch the graph of 𝑦 = 𝑓(𝑥).
[2]
−1
−1
Obtain an expression for 𝑓 (𝑥), stating both the domain and range of 𝑓 .
[4]
(J2017/P11/Q3)
(a)
Prove the identity
1+𝑐𝑜𝑠𝜃
𝑠𝑖𝑛𝜃
(b) Hence solve the equation
𝑠𝑖𝑛𝜃
2
+ 1+𝑐𝑜𝑠𝜃 ≡ 𝑠𝑖𝑛𝜃
1+𝑐𝑜𝑠𝜃
𝑠𝑖𝑛𝜃
𝑠𝑖𝑛𝜃
[3]
3
+ 1+𝑐𝑜𝑠𝜃 = 𝑠𝑖𝑛𝜃 for 0 ≤ 𝜃 ≤ 360
[3]
MATH IN FOCUS WITH RAJA NADEEM AMIN 03215862394
Q20.
(J2017/P11/Q3)
The equation of a curve is 𝑦 = 2 𝑐𝑜𝑠 𝑥.
(i) Sketch the graph of 𝑦 = 2 𝑐𝑜𝑠 𝑥 for −𝜋 ≤ 𝑥 ≤ 𝜋, stating the coordinates of the point of
intersection with the y-axis.
[2]
𝜋
Points P and Q lie on the curve and have x-coordinates of 3 and 𝜋 respectively.
(ii) Find the length of 𝑃𝑄 correct to 1 decimal place.
[2]
The line through 𝑃 and 𝑄 meets the 𝑥 − 𝑎𝑥𝑖𝑠 at 𝐻(ℎ, 0) and the 𝑦 − 𝑎𝑥𝑖𝑠 at 𝐾(0, 𝑘).
5𝜋
(iii) Show that ℎ = 9 and find the value of 𝑘.
Q21.
[3]
(J2017/P12/Q3)
(i)
Prove the identity
2
1
1−𝑠𝑖𝑛𝜃
(𝑐𝑜𝑠𝜃 − 𝑡𝑎𝑛 𝜃) ≡ 1+𝑠𝑖𝑛𝜃
(ii)
[3]
Hence solve the equation
2
1
1
(𝑐𝑜𝑠𝜃 − 𝑡𝑎𝑛 𝜃) = 2 for 00 ≤ 𝜃 ≤ 3600
Q22.
Q23.
[3]
(J2017/P13/Q5)
2𝑠𝑖𝑛𝜃+𝑐𝑜𝑠𝜃
(i)
Show that the equation 𝑠𝑖𝑛𝜃+𝑐𝑜𝑠𝜃 = 2𝑡𝑎𝑛𝜃 may be expressed as cos 2 𝜃 = 2 sin2 𝜃
(ii)
(ii) Hence solve the equation 𝑠𝑖𝑛𝜃+𝑐𝑜𝑠𝜃 = 2𝑡𝑎𝑛𝜃 for 00 < 𝜃 < 1800 .
2𝑠𝑖𝑛𝜃+𝑐𝑜𝑠𝜃
Prove the identity
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
(viii)
1 − cos 2 𝑥 ≡ sin2 𝑥
(1 − sin2 𝜃)𝑡𝑎𝑛𝜃 ≡ 𝑐𝑜𝑠𝜃𝑠𝑖𝑛𝜃
1
sin2 𝜃
cos2 𝜃
− sin2 𝜃 ≡ 1
1
tan2 𝜃 ≡ cos2 𝜃 − 1
sin2 𝜃 −3 cos2 𝜃+1
sin2 𝜃−cos2 𝜃
1
1
cos2 𝜃
≡2
1
+ sin2 𝜃 ≡ cos2 𝜃 sin2 𝜃
𝑐𝑜𝑠𝜃
1
𝑡𝑎𝑛𝜃 + 𝑠𝑖𝑛𝜃 ≡ 𝑠𝑖𝑛𝜃𝑐𝑜𝑠𝜃
1
1+𝑠𝑖𝑛𝜃
1
2
+ 1−𝑠𝑖𝑛𝜃 ≡ cos2 𝜃
[3]
[3]
MATH IN FOCUS WITH RAJA NADEEM AMIN 03215862394
1−tan2 𝑥
(ix)
Prove the identity 1+tan2 𝑥 ≡ 1 − 2 sin2 𝑥
(x)
Prove the identity 𝑐𝑜𝑠𝑥 + 1+𝑠𝑖𝑛𝑥 ≡ 𝑐𝑜𝑠𝑥
(xi)
Prove the identity 1−𝑠𝑖𝑛𝑥 − 1+𝑠𝑖𝑛𝑥 ≡ 2 tan2 𝑥
(xii)
Prove the identity
1+𝑠𝑖𝑛𝑥
𝑐𝑜𝑠𝑥
𝑠𝑖𝑛𝑥
𝑠𝑖𝑛𝑥
2
(
1+𝑐𝑜𝑠𝜃
(xiii)
Prove the identity
(xiv)
Prove the identity 𝑡𝑎𝑛𝜃(
+
𝑠𝑖𝑛𝜃
𝑠𝑖𝑛𝜃
1+𝑐𝑜𝑠𝜃
𝑐𝑜𝑠𝜃
1−𝑠𝑖𝑛𝜃 )
≡
2
1
1 − 𝑠𝑖𝑛𝜃
− 𝑡𝑎𝑛 𝜃) ≡
𝑐𝑜𝑠𝜃
1 + 𝑠𝑖𝑛𝜃
2
𝑠𝑖𝑛𝜃
1
≡ 1 + 𝑠𝑖𝑛𝜃