MAS110 Problems for Chapter 3: Trigonometry
1. Prove that sin θ = √
tan θ
1 + tan2 θ
and cos θ = √
1
1 + tan2 θ
, if −π/2 < θ < π/2.
2. Consider a circle with centre O and points A, B, C on the circumference. By considering the
isosceles triangles OAB, OAC and OBC, and by considering the sums of angles in the triangles
ABC and OBC, show that ∠BOC = 2∠BAC. For ease, draw your picture so that O is inside
the triangle ABC.
3. Let C be the circle x2 + y 2 = 1. Let L be the straight line through the points (−1, 0) and (0, t).
What is the equation of L? At what point other than (−1, 0) does it intersect C? Express the
coordinates (x, y) of this point in terms of t.
4. Show that
2 tan θ
1 − tan2 θ
and sin 2θ =
.
2
1 + tan θ
1 + tan2 θ
What has this got to do with the previous two questions? [Hint: let θ be an angle between L and
the x-axis.]
cos 2θ =
5. Write down the formulas for sin(θ + φ) and cos(θ + φ) in terms of sin θ, cos θ, sin φ and cos φ.
(i) Find formulas for cos(2θ) and sin(2θ).
(ii) Applying the addition formulas to 2θ and θ, prove that cos(3θ) = cos3 θ − 3 cos θ sin2 θ
and sin(3θ) = 3 sin θ cos2 θ − sin3 θ. Prove also that cos(5θ) = cos5 θ − 10 cos3 θ sin2 θ +
5 cos θ sin4 θ and sin(5θ) = 5 sin θ cos4 θ − 10 sin3 θ cos2 θ + sin5 θ. (Remark: in case you are
thinking there must be an easier way, we shall see one later.)
(iii) Express cos(3θ) and cos(5θ) as polynomials in cos θ. Write out sin(3θ) and sin(5θ) as
polynomials in sin θ. You should find that sin(5θ) = 16 sin5 θ − 20 sin3 θ + 5 sin θ.
(iv) Solve sin(5θ) = 0 for − π2 ≤ θ ≤ π2 . Also, find the roots of 16x5 − 20x3 + 5x = 0. Hence
derive explicit expressions for sin(π/5) and sin(2π/5).
6. Suppose 3π/2 ≤ θ ≤ 2π and cos θ = 53 . Evaluate sin θ, sin 2θ, cos 2θ and cos 3θ.
7. The n-th Tchebychev polynomial Tn (x) is defined by setting T0 (x) := 1, T1 (x) := x and using
the recurrence Tn (x) = 2xTn−1 (x) − Tn−2 (x) when n ≥ 2.
(i) Write down T2 (x), T3 (x), T4 (x) and T5 (x) explicitly. What is the degree of Tn (x)?
(ii) Express cos(2θ) and cos(4θ) as polynomials in cos θ. Either recall from Question 5, or do
the same for cos(3θ) and cos(5θ).
(iii) What do you notice? Make a claim, and prove it.
1
8. Show that the area of a regular n-gon inscribed in a circle of radius 1 is n2 sin 2π
. (Use the centre
n
to break up the polygon into n congruent isosceles triangles.) What happens when we make n
large?
If we use a square, then a regular octagon, then a regular hexadecagon, and so on, we obtain the
sequence of areas
π
an := 2n sin n , n = 1, 2, 3, . . . .
2
π
This can be used to make practical approximations as follows. Set bn := cos n , n = 1, 2, 3, . . ..
2
Show that, for all n ≥ 1,
r
bn + 1
an
and an+1 =
.
bn+1 =
2
bn+1
Now use these relations to calculate a10 (or even a little beyond!).
9. Show the area of a triangle ABC is given by
is the semi-perimeter of the triangle.
p
s(s − a)(s − b)(s − c) where s := (a + b + c)/2
Hint: The area is 12 bc sin A. Derive expressions for 1 ± cos A starting with cos A =
p
simplify as far as possible. Now calculate sin A = (1 + cos A)(1 − cos A).
b2 +c2 −a2
2bc
and
√
2
2
10. Show that if b 6=
√ 0 then a cos θ + b sin θ = R sin(θ + α), where R = a + b and tan(α) = a/b.
Hence express 3 cos θ − sin θ in the form R sin(θ + α). Is there an analogous formula involving
an expression of the form R cos(θ + α)?
11. How are x1 and x2 related if sin x1 = sin x2 ? What if sin x1 = cos x2 ? Find all possible values of
θ satisfying cos θ = sin 4θ in the range 0 < θ < π.
You should find that θ = π/10 satisfies cos θ = sin 4θ. Let s = sin(π/10). By expressing sin 4θ
2
2
in terms of sin θ, show that s satisfies the relation 1 = 4s(1
√ − 2s ). Factoring 1 − 4s(1 − 2s ) as
(2s − 1) × (a quadratic in s), deduce that sin(π/10) = ( 5 − 1)/4.
12. Write down the formulas for sin(θ + φ) and cos(θ + φ) in terms of sin θ, cos(θ), sin(φ) and cos(φ).
Deduce formulas for sin(θ − φ) and cos(θ − φ).
(i) Recall, with justification, the values of sin(π/3), cos(π/3), sin(π/4) and cos(π/4).
(ii) Deduce the exact values of sin(π/12) and cos(π/12).
(iii) Give the deduction, from the addition formulas, of formulas for sin2 θ and cos2 θ. Now use
them to obtain exact values for sin(π/24) and cos(π/24).
(iv) Using what you found in Question 5, how about sin(π/20) and cos(π/20)?
Techniques including these were used by Ptolemy (100–178 A.D.) to produce tables of sines for all
multiples of half a degree. Ulugh Beg (1394–1449, beheaded by his eldest son) produced tables
for every minute (sixtieth of a degree) to about 9 decimal places. We shall see an easier way later.
13. Using the addition formulas for sin and cos, find a formula for tan(θ + φ) in terms of tan(θ) and
tan(φ). Deduce a formula for tan(2θ). If tan A = 1/5 and tan B = 1/239, use your formulas to
2
calculate tan 2A, tan 4A and tan(4A − B). Deduce that
π/4 = 4 tan−1 (1/5) − tan−1 (1/239).
This was discovered by John Machin (1680–1751), and is useful for approximating π (as we shall
see later).
14. Write down the addition formula for tan(x + y), and show that
π θ
tan
−
= sec θ − tan θ.
4 2
(i) Write down the values of tan(π/3) and tan(π/4). By substituting in θ = π/4 in the above
formula, find the value of tan(π/8).
√
(ii) Show that tan(π/12) = 2 − 3 and
√
√
3+ 2−1
√
tan(11π/24) = √
.
3− 6+1
[Hint: use the addition formula for tan(x + y)
choices of x and y.] Simplify
√ with√convenient
√
the right hand side to obtain tan 11
π
=
2
+
2
+
3
+
6.
24
p
√
√
√
√
√
√
(iii) Show that tan(π/48) = 16 + 10 2 + 8 3 + 6 6 − (2 + 2 + 3 + 6).
3 tan θ − tan3 θ
.
1 − 3 tan2 θ
(i) Evaluate tan 3θ if tan θ = 1/2.
√
(ii) Suppose π/4 ≤ θ < π/2 and tan 3θ = 11/2. Show that tan θ = 8 + 5 3.
15. Show that tan 3θ =
3