Trigonometry

1. Trigonometry — The Core Idea

Trigonometry is the study of relationships between angles and lengths. At its simplest level, it connects the sides of a right-angled triangle. At A Level, the subject develops into a much broader system involving functions, identities, equations, graphs, radians and non-right-angled triangles.

Angle notation

The usual convention is to use capital letters for angles and lowercase letters for their opposite sides:

Degrees and radians

Degrees divide a complete revolution into 360 equal parts. Radians measure an angle using the arc length of a circle.

2. Basic Trigonometric Ratios

In a right-angled triangle, the three fundamental ratios are sine, cosine and tangent.

Finding a side

If the angle and another side are known, choose the ratio that contains the required side and the known side.

Worked Example

A right-angled triangle has an angle of 37° and a hypotenuse of 12 cm. Find the side opposite the angle.

Finding an angle

Use an inverse trigonometric function when a side ratio is known.

3. Exact Trigonometric Values

The standard angles 0°, 30°, 45°, 60° and 90° produce exact values that are fundamental to A Level trigonometry.

Why do these values occur?

The values can be derived from special triangles rather than memorised as isolated facts.

45°–45°–90° triangle

If the equal perpendicular sides have length 1, Pythagoras gives a hypotenuse of √2.

30°–60°–90° triangle

The side lengths can be represented as 1, √3 and 2.

4. The Unit Circle and Trigonometric Signs

The unit circle gives a powerful way to understand trigonometric functions for angles beyond the first quadrant.

A point on the unit circle corresponding to angle θ has coordinates:

Therefore the x-coordinate represents cosine and the y-coordinate represents sine.

Signs in the four quadrants

CAST

CAST is a memory aid. The actual mathematics comes from the signs of the coordinates and tan θ = sin θ / cos θ.

5. Reciprocal Trigonometric Functions

The reciprocal functions are cosecant, secant and cotangent.

Important identities

Domain restrictions

Since a reciprocal cannot have zero as its denominator:

6. Trigonometric Identities

An identity is an equation that is true for every value for which both sides are defined.

The fundamental identity

Deriving the tangent identity

Divide the fundamental identity by cos²θ:

Deriving the cotangent identity

Dividing the fundamental identity by sin²θ gives:

Identity manipulation strategy

1. Usually work on one side only.
2. Convert tan into sin/cos when useful.
3. Use sin²θ + cos²θ = 1.
4. Look for common denominators.
5. Factor where possible.
6. Do not divide by an expression without considering where it may be zero.

Worked Example

Show that:

7. Solving Trigonometric Equations

A trigonometric equation may have several solutions within a specified interval. The principal inverse-trigonometric answer is usually only the starting point.

Example 1 — sine

The reference angle is 30°. Sine is positive in Quadrants I and II.

Example 2 — cosine

The reference angle is 30°. Cosine is negative in Quadrants II and III.

Example 3 — tangent

The reference angle is 45°. Tangent is positive in Quadrants I and III.

Radian form

8. Quadratic Trigonometric Equations

Many harder trigonometric equations are actually algebraic equations in disguise. Treat sin x, cos x or tan x as an algebraic variable.

Worked Example

Solve:

Factorise:

Therefore:

The first equation gives x = 30°, 150°. The second gives x = 90°.

Key method

1. Rewrite the equation as a polynomial in one trig function.
2. Factorise or use the quadratic formula.
3. Solve each resulting trigonometric equation.
4. Check the complete interval.

9. Transforming Trigonometric Equations

Some equations cannot be solved immediately because they contain different trigonometric functions. Identities allow them to be transformed into a solvable form.

Example

Recognise the double-angle identity:

Therefore:

The important point is that solving for 2x requires careful treatment of the interval. If:

then:

Hence:

10. The Sine Rule

The Sine Rule connects each side of a triangle with the sine of its opposite angle.

When is it useful?

• A known side and its opposite angle are available.
• You need another side when its opposite angle is known.
• You need an angle when its opposite side is known.
• It is particularly important in the ambiguous SSA case.

Worked Example

A = 30°, B = 60°, a = 5. Find b.

11. The Ambiguous Case of the Sine Rule

When two sides and a non-included angle are known, the Sine Rule can produce two possible angles.

Suppose:

If you calculate B using sin⁻¹, remember:

Therefore both B and 180° − B may need to be checked.

Example

But:

Both must be checked against the triangle condition A + B + C = 180°.

12. The Cosine Rule

The Cosine Rule generalises Pythagoras to any triangle.

Cyclic versions are:

When to use it

• Two sides and the included angle are known.
• All three sides are known and an angle is required.

Worked Example

Let b = 5, c = 7 and A = 60°. Find a.

Finding an angle

Rearrange:

13. Area of a Triangle

If two sides and the included angle are known, the area is:

The same relationship can be written cyclically:

Why does this work?

If side b is used as the base and side a is the sloping side, the perpendicular height is a sin C.

Worked Example

Find the area when a = 8, b = 5 and C = 30°.

14. Compound-Angle Identities

Compound-angle identities express trigonometric functions of sums and differences in terms of the individual angles.

Memory pattern

For cosine, the sign between the two products is opposite to the sign inside the original angle.

Worked Example

Find the exact value of sin 75°.

Write:

15. Double-Angle Identities

Set A = B in the compound-angle identities.

Choosing the useful form

If the expression contains cos²A, use:

If it contains sin²A, use:

If both sin²A and cos²A appear, the form cos²A − sin²A is often most convenient.

Worked Example

Given sin A = 3/5 and cos A = 4/5, find sin 2A and cos 2A.

16. The Graphs of sin x, cos x and tan x

y = sin x

Important points over one cycle:

y = cos x

y = tan x

Tangent has vertical asymptotes wherever cos x = 0.

17. Transformations of Trigonometric Graphs

The general sinusoidal form is:

Effect of a

The magnitude |a| controls the amplitude.

If a is negative, the graph is reflected in the x-axis.

Effect of b

The period becomes:

In degrees:

Effect of d

d moves the graph vertically.

Range

Worked Example

• Amplitude = 3
• Period = π
• Midline = y = −1
• Maximum = 2
• Minimum = −4
• Range: −4 ≤ y ≤ 2

18. Horizontal Translation and Phase

Expressions of the form:

have a horizontal shift of h.

Example

• Amplitude = 2
• Period = 2π/3
• Shift right by π/6
• Midline = y = 1
• Range = −1 to 3

19. Proving Trigonometric Identities

Proof questions test whether you can manipulate expressions systematically rather than simply calculate numerical values.

Worked Proof

Prove:

Start with the left-hand side:

Use a common denominator:

Use the fundamental identity:

Split the reciprocal:

20. Radians

A radian is defined using the relationship between arc length and radius.

where θ is measured in radians, s is the arc length and r is the radius.

Arc length

Sector area

Worked Example — Arc Length

A circle has radius 8 cm and central angle π/3 radians.

Worked Example — Sector Area

21. Why Radians Matter

Radians are not merely an alternative angle unit. They make many trigonometric relationships take their simplest mathematical form.

For example, when θ is small and measured in radians:

This approximation is not generally valid when θ is measured in degrees.

Similarly:

for sufficiently small θ in radians.

22. Periodicity and General Solutions

Trigonometric functions repeat their values periodically.

General sine solution

If sin x = sin α, the general solutions can be written:

General cosine solution

General tangent solution

In examination questions, the interval usually determines which members of these infinite families are required.

23. Expressing a sin x + b cos x in R Form

Consider:

Expand the right-hand side:

Compare coefficients:

Squaring and adding gives:

Then:

Worked Example

Express:

in the form R sin(x + α).

Why R-form matters

Once an expression is written as R sin(x + α), its maximum and minimum become immediately visible:

24. Maximum and Minimum Values

Since:

it follows that:

Example

Since −1 ≤ cos x ≤ 1:

Therefore the minimum value is 4 and the maximum value is 10.

25. Parameter Problems

Trigonometric questions often introduce an unknown constant and ask you to determine its possible values.

Example — range

Suppose:

The range is:

If the question says the minimum value is 7:

Parameter + number of roots

Questions may ask how many solutions an equation has for a particular interval. A graph or range analysis can be more efficient than solving every equation directly.

26. Choosing the Correct Triangle Method

Use SOHCAHTOA when:

The triangle is right-angled and the relevant sides/angle are known.

Use the Sine Rule when:

You know an opposite side-angle pair and need another side or angle.

Use the Cosine Rule when:

You know two sides and their included angle, or all three sides.

Use the area formula when:

Two sides and the included angle are known and the area is needed.

27. Master Strategy for Trigonometric Equations

1. Check the interval. Write it down before calculating.
2. Check the calculator mode. Determine whether the question uses degrees or radians.
3. Simplify first. Factorise or use identities before pressing buttons.
4. Find the reference angle. Use the positive principal value.
5. Use signs. Determine the relevant quadrants using CAST or the unit circle.
6. Generate every solution. Do not stop after inverse trig gives one answer.
7. Check the interval. Remove values outside the required range.

28. Common Trigonometry Mistakes

1. Degree/radian confusion

The calculator may return a completely different answer if it is in the wrong mode.

2. Confusing sin⁻¹ with cosec

sin⁻¹x means inverse sine, whereas cosec x = 1/sin x.

3. Missing the second solution

Inverse trig functions return only a principal value. The complete interval must still be considered.

4. Wrong CAST quadrant

Remember that tangent is positive in Quadrants I and III, not Quadrants I and IV.

5. Wrong angle in the Sine Rule

Each side must correspond to its opposite angle.

6. Wrong angle in the area formula

1/2ab sin C requires C to be the included angle between a and b.

7. Using the Cosine Rule incorrectly

The angle must be opposite the side being squared in:

8. Treating an identity as an equation

An identity is true for all valid values; it is not solved for a small collection of answers.

9. Rounding too early

Keep exact values or calculator precision until the final step.

10. Forgetting transformed intervals

If you replace x with 2x, the interval for 2x changes as well.

29. Trigonometry Exam Checklist

• Is my calculator in the correct mode?
• Did I use degrees or radians correctly?
• Did I identify the correct trigonometric relationship?
• Did I match sides with their opposite angles?
• Did I use the included angle for the area formula?
• Did I consider every quadrant?
• Did I check the entire interval?
• Did I preserve exact values when required?
• Did I avoid premature rounding?
• Does my answer make geometric sense?

30. Trigonometry Formula Atlas

Basic ratios

Fundamental identities

Reciprocals

Compound angles

Double angles

Triangle geometry

Radians

Sinusoidal graphs

R-form

31. Practice Questions — Foundation

Attempt each question before opening the solution.

Question 1 — Basic ratio

In a right-angled triangle, the opposite side is 8 cm and the hypotenuse is 10 cm. Find θ.

Question 2 — Exact identity

Given sin θ = 5/13 and θ is acute, find cos θ.

Question 3 — Equation

Solve sin θ = √3/2 for 0° ≤ θ ≤ 360°.

Question 4 — Cosine Rule

Two sides are 6 and 8 and the included angle is 60°. Find the third side exactly.

Question 5 — Area

Find the area of a triangle with sides 10 cm and 6 cm and included angle 30°.

32. Practice Questions — Intermediate

Question 6 — Quadratic trig equation

Solve: 2sin²x − 3sinx + 1 = 0, for 0° ≤ x ≤ 360°.

Question 7 — Compound angle

Find the exact value of cos 75°.

Question 8 — Double angle

Given cos θ = 5/13 and θ is acute, find cos 2θ.

Question 9 — Radians

A sector has radius 9 cm and angle 2π/3 radians. Find its area.

Question 10 — Graph range

Find the range of y = 4 − 3sin x.

33. Practice Questions — Hard / Multi-Step

Question 11 — Identity + equation

Solve: 1 − 2sin²x = 0, for 0° ≤ x ≤ 180°.

Question 12 — R-form

Express 5sinx + 12cosx in the form R sin(x + α), where R > 0 and 0 < α < 90°.

Question 13 — Maximum value

Find the maximum value of 8 + 6sin x − 8cos x.

Question 14 — Identity proof

Prove: (1 − cosx)/sinx = sinx/(1 + cosx).

Question 15 — Mixed triangle reasoning

In triangle ABC, a = 7, b = 9 and C = 60°. Find the area exactly.

34. Deep Reasoning — A Level Challenge

Challenge 1

Given:

Find θ for 0° ≤ θ ≤ 360°.

Challenge 2

Show that the maximum value of a sinx + b cosx is √(a² + b²).

Challenge 3

Solve: 3cos²x − 2cosx − 1 = 0 for 0° ≤ x ≤ 360°.

35. Trigonometry Master Checklist

You should be able to confidently:

• Use SOHCAHTOA in right-angled triangles.
• Convert between degrees and radians.
• Recall exact standard-angle values.
• Use the unit-circle interpretation of sine and cosine.
• Determine signs in all four quadrants.
• Use reciprocal trigonometric functions.
• Manipulate fundamental trigonometric identities.
• Solve sine, cosine and tangent equations.
• Solve quadratic trigonometric equations.
• Use compound-angle identities.
• Use double-angle identities.
• Use the Sine Rule.
• Handle the ambiguous Sine Rule case.
• Use the Cosine Rule.
• Find the area of a triangle using 1/2ab sinC.
• Use radians for arc length and sector area.
• Understand sine, cosine and tangent graphs.
• Determine amplitude, period, range and midline.
• Handle graph transformations.
• Prove trigonometric identities.
• Convert a sinx + b cosx into R-form.
• Find maximum and minimum values.
• Solve parameter-based trigonometric problems.
• Handle multi-step examination problems.