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Additional Mathematics Synthesis Guide 3: Trigonometric Functions, Identities, Equations and Geometry

BUKIT TIMAH TUTOR · ADDITIONAL MATHEMATICS · SYNTHESIS GUIDE 3

Trigonometry becomes coherent when sine, cosine and tangent are treated as functions with geometry, graphs, algebra and periodicity all describing the same object.

A-Math trigonometry can feel like several unrelated topics: identities, equations, graph transformations, exact values, radians and triangle geometry. The deeper structure is simpler. An angle selects a position or ratio. The trigonometric function returns a value. Periodicity creates repeated values. Identities describe relationships that remain true whenever both sides are defined. Equations ask which angles produce a required value.

Angle → function value → graph → periodic repetition → identity → equation → geometric interpretation.

1. From right-triangle ratios to functions

In an acute right triangle, sine, cosine and tangent begin as ratios of sides. Additional Mathematics extends them beyond one triangle. We treat sin x, cos x and tan x as functions defined for broad families of angles. This shift matters because a function can be graphed, transformed, composed with algebra and solved over intervals.

The unit-circle viewpoint is especially useful. For a point on the unit circle at angle θ from the positive x-axis, its coordinates are (cos θ, sin θ). This makes signs by quadrant, periodicity and identities much easier to understand.

2. The signs come from coordinates, not a mnemonic alone

  • Quadrant I: sin, cos and tan are positive.
  • Quadrant II: sin positive, cos negative, tan negative.
  • Quadrant III: sin negative, cos negative, tan positive.
  • Quadrant IV: sin negative, cos positive, tan negative.

A mnemonic can help recall the pattern, but coordinates explain it. Tangent has the sign of sin θ / cos θ, so it is positive when sine and cosine have the same sign and negative when they differ.

3. Periodicity is why trigonometric equations have multiple answers

Sine and cosine repeat every 360° or 2π radians. Tangent repeats every 180° or π radians. Therefore solving a trigonometric equation is not complete when one calculator angle is found. The student must locate all relevant angles in the requested interval.

For example, sin x = 1/2 has one principal calculator value, 30°, but on 0° ≤ x ≤ 360° the solutions are 30° and 150°.

4. Graphs show the repetition directly

For y = a sin(bx) + c or y = a cos(bx) + c in degree mode:

  • |a| controls amplitude.
  • The period is 360°/|b|.
  • c shifts the midline vertically.
  • A negative a reflects the graph vertically.

For tangent, the period is 180°/|b| and vertical asymptotes matter because tan x is undefined when cos x = 0.

5. Worked Example 1 — read a transformed sine graph

For y = 3 sin(2x) − 1, state the amplitude, period and midline.

Amplitude = 3. Period = 360°/2 = 180°. Midline is y = −1. Therefore the maximum value is 2 and the minimum is −4.

6. Identities are equations that remain true across their domain

The basic identity sin2x + cos2x = 1 is not an equation to solve for x. It is a relationship that is true for every x for which the expressions are defined.

Other useful relationships include:

  • tan x = sin x / cos x
  • 1 − cos 2x = 2 sin2x
  • 1 + cos 2x = 2 cos2x
  • sin 2x = 2 sin x cos x

The goal in an identity proof is to transform one side using known valid relationships until it becomes the other. It is usually safer to work on the more complicated side rather than manipulating both sides simultaneously and losing track of implication.

7. Worked Example 2 — prove an identity structurally

Show that

(1 − cos 2x)/sin 2x = tan x

where both sides are defined.

Use double-angle identities:

1 − cos 2x = 2 sin2x and sin 2x = 2 sin x cos x.

Then

(2 sin2x)/(2 sin x cos x) = sin x/cos x = tan x.

The proof works because the numerator and denominator were rewritten into compatible factors before cancellation.

8. Exact values preserve structure

Exact trigonometric values matter because later algebra may require symbolic cancellation or comparison. A decimal approximation can hide that structure.

Using the angle-addition formula:

sin 75° = sin(45° + 30°)

= sin45° cos30° + cos45° sin30°

= (√2/2)(√3/2) + (√2/2)(1/2)

= (√6 + √2)/4.

9. Equations often become algebra in a trigonometric variable

Consider 2cos2x − 3cos x + 1 = 0. Let c = cos x. Then

2c2 − 3c + 1 = 0

(2c − 1)(c − 1) = 0.

So cos x = 1/2 or cos x = 1. The problem is now split into ordinary trigonometric equations.

10. Worked Example 3 — solve all solutions in an interval

Solve 2cos2x − 3cos x + 1 = 0 for 0° ≤ x ≤ 360°.

From the factorisation, cos x = 1/2 or cos x = 1.

For cos x = 1/2, x = 60°, 300°. For cos x = 1, x = 0°, 360° within the stated inclusive interval.

Therefore x = 0°, 60°, 300°, 360°.

11. Squaring can create extra solutions

When a trigonometric equation is squared to remove a sign or radical, the transformed equation may admit solutions that the original equation did not. The student must substitute candidates back into the original equation. This is not a special trigonometry rule; it is a general algebra rule about transformations that are not reversible without conditions.

12. Radians make geometry and calculus fit together

Radians measure angle using arc length. If an arc of length s lies on a circle of radius r, then θ = s/r when θ is measured in radians. Therefore

s = rθ.

The corresponding sector area is (1/2)r2θ. These formulas are compact because the radian measure already contains the ratio between arc length and radius.

13. Worked Example 4 — radians and sector geometry

A sector has radius 6 cm and angle 1.4 radians. Find its arc length and area.

Arc length: s = rθ = 6(1.4) = 8.4 cm.

Area: (1/2)r2θ = (1/2)(36)(1.4) = 25.2 cm2.

14. Triangle geometry is another representation of trigonometric relationships

In non-right triangles, the sine rule and cosine rule extend the same angle-side relationships beyond SOH-CAH-TOA. Area can be found from two sides and the included angle using A = (1/2)ab sin C.

For sides 7 and 9 with included angle 60°, the area is

(1/2)(7)(9)sin60° = 63√3/4.

The same sine function that appears on a periodic graph therefore also controls area in a triangle.

15. Graphs can check equations

If an equation asks where sin x = 0.6, imagine or sketch the horizontal line y = 0.6 across y = sin x. Each intersection corresponds to a solution. This gives a powerful verification method: the number and approximate positions of algebraic solutions should agree with the graph.

16. Common failure patterns

  • Taking only the calculator’s first angle. Periodicity usually creates additional solutions.
  • Mixing degree and radian mode. Correct mathematics in the wrong calculator state produces a wrong numerical answer.
  • Using an identity as though it were an equation to solve.
  • Manipulating both sides of an identity proof until the logical direction becomes unclear.
  • Replacing exact values with decimals too early.
  • Dividing by a trigonometric expression that could be zero. This can discard valid solutions.
  • Forgetting interval endpoints.
  • Ignoring domains of tangent and other quotients.

17. A reliable equation-solving routine

  1. Confirm degree or radian mode.
  2. State the required interval.
  3. Simplify the equation using identities or algebra.
  4. Factor before dividing whenever possible.
  5. Solve the basic trigonometric equation.
  6. Use quadrant signs and periodicity to generate all solutions.
  7. Check endpoints and excluded values.
  8. Substitute back if any non-reversible operation such as squaring was used.

18. Practice set

  1. State the amplitude and period of y = 4cos(3x) + 2 in degrees.
  2. Solve sin x = √3/2 for 0° ≤ x ≤ 360°.
  3. Solve tan x = 1 for 0° ≤ x < 360°.
  4. Solve 2sin2x − sin x − 1 = 0 for 0° ≤ x ≤ 360°.
  5. Show that (1 − cos 2x)/sin 2x = tan x where defined.
  6. Find the exact value of cos 75°.
  7. Convert 150° to radians.
  8. Find the arc length for r = 5 cm and θ = 2.2 radians.
  9. Find the sector area for r = 4 cm and θ = 1.5 radians.
  10. Find the area of a triangle with sides 8 and 11 enclosing 30°.
  11. Explain why dividing an equation by sin x can lose solutions.
  12. Explain why an equation can have several solutions even when a calculator displays one inverse-trig value.

Answers

  1. Amplitude 4; period 120°.
  2. 60°, 120°.
  3. 45°, 225°.
  4. sin x = 1 or sin x = −1/2, so x = 90°, 210°, 330°.
  5. Use 1 − cos2x = 2sin2x and sin2x = 2sinx cosx, then simplify.
  6. (√6 − √2)/4.
  7. 5π/6.
  8. 11 cm.
  9. 12 cm2.
  10. 22 square units.
  11. Because sin x may be zero; division by zero is not valid, and those zero cases may satisfy the original equation.
  12. Inverse trig returns a principal value, while periodic functions repeat the same output at other angles.

19. What mastery looks like

A strong A-Math learner can read a trig graph, connect signs to quadrants, move between degrees and radians, distinguish an identity from an equation, preserve exact values, solve algebraically structured trigonometric equations, generate all solutions in an interval and use geometry or a graph to check the result.

The transfer test is not another nearly identical equation. Change the interval. Change the mode. Hide the quadratic structure inside sin x or cos x. Present the same relationship as a graph or a triangle. If the student can still locate the invariant structure, trigonometry is becoming one connected system rather than a collection of tricks.

Continue through the BTT Additional Mathematics system