Trigonometry in Secondary 4 Additional Mathematics is not mainly about remembering more formulas. It is about recognising structure, preserving exact relationships and controlling all valid solutions. The same sine, cosine and tangent functions that began with triangles become full algebraic objects: they have graphs, periods, identities, transformations, equations, inverse relationships and connections to calculus.
This Secondary 4 Additional Mathematics Learning Guide develops trigonometry as one connected system. It belongs to the Additional Mathematics Learning Hub and connects directly to calculus, functions, algebra and coordinate reasoning.
Angle → function value → graph → identity → equation → complete solution set → verification.
A trigonometric ratio becomes a trigonometric function
In right-triangle work, sine, cosine and tangent begin as ratios. In Additional Mathematics, the functions must operate beyond acute triangles. Angles can be obtuse, reflex, negative or extend through several revolutions. The values repeat according to periodic structure.
This means sin x=1/2 does not have one answer. It has many possible angles, and the interval supplied by the question decides which solutions are relevant.
| Function | Period |
|---|---|
| sin x | 360° or 2π |
| cos x | 360° or 2π |
| tan x | 180° or π |
Degrees and radians
A full revolution is 360° or 2π radians, so 180°=π radians. Hence 60°=π/3, 45°=π/4, 30°=π/6 and 270°=3π/2.
The calculator mode must match the angle unit. But radians are not a different kind of angle; they are another unit for measuring the same rotation.
Exact standard-angle values
| Angle | sin | cos | tan |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | 1/2 | √3/2 | 1/√3 |
| 45° | √2/2 | √2/2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
| 90° | 1 | 0 | undefined |
These values form reference points. Other quadrant values can be determined from reference angles and signs rather than memorised individually.
Quadrant reasoning
For 150°, the reference angle is 30°. The sine magnitude is 1/2. Because 150° lies in Quadrant II, sine is positive, cosine negative and tangent negative.
Worked Example 1 | Exact value in another quadrant
cos150° has reference angle 30°. Cosine is negative in Quadrant II, so cos150°=−√3/2.
Graphs are the operating map
The sine, cosine and tangent graphs explain periodicity, signs, maxima, minima, zeros and asymptotes. They show why equations can have several solutions in one interval.
- sin x has amplitude 1 and period 360° or 2π.
- cos x has the same amplitude and period but begins at a maximum when x=0.
- tan x has period 180° or π and vertical asymptotes where cos x=0.
Transformations of trigonometric graphs
For y=a sin(bx+c)+d or the cosine equivalent, |a| controls amplitude, b controls period, c contributes to horizontal phase shift and d moves the midline vertically.
For y=3sin(2x)+1 in degrees, the amplitude is 3, the midline is y=1 and the period is 180°.
Inside changes horizontal behaviour. Outside changes vertical behaviour.
Worked Example 2 | Read a transformed sine graph
For y=2sin(3x)−4 in degrees, amplitude=2, period=120°, midline y=−4, maximum=−2 and minimum=−6.
The fundamental identity
sin²x+cos²x=1.
This identity allows one function to be replaced with the other. Rearrangements include sin²x=1−cos²x and cos²x=1−sin²x.
Tangent connects sine and cosine
tan x=sin x/cos x.
This lets tangent be rewritten entirely in sine and cosine and explains why tangent is undefined whenever cos x=0.
Proving identities
An identity is true for every value in its valid domain. A proof normally begins with one side and transforms it until it becomes the other side.
- replace tan x with sin x/cos x;
- replace sin²x+cos²x with 1;
- factorise;
- combine fractions;
- work from the structurally more complicated side.
Worked Example 3 | Simple identity
(1−cos²x)/sin x = sin²x/sin x = sin x, where the expressions are defined.
Worked Example 4 | Identity involving tangent
tan x cos x=(sin x/cos x)cos x=sin x.
Trigonometric equations
A calculator usually gives a principal angle. The interval determines the full solution set. A reliable sequence is:
- Isolate one trigonometric function.
- Find a reference or principal angle.
- Identify quadrants with the required sign.
- Generate every solution in the interval.
- Check endpoints and exclusions.
Worked Example 5 | Sine equation
Solve 2sin x=1 for 0°≤x≤360°. Then sin x=1/2. Sine is positive in Quadrants I and II, so x=30°,150°.
Worked Example 6 | Cosine equation
cos x=−√3/2 has reference angle 30°. Cosine is negative in Quadrants II and III, so x=150°,210°.
Worked Example 7 | Tangent equation
tan x=1 has reference angle 45° and tangent is positive in Quadrants I and III, so x=45°,225°.
Transformed angles require transformed search intervals
For sin2x=1/2 with 0°≤x≤360°, the inside angle 2x lies in 0°≤2x≤720°. Search the full doubled interval before dividing solutions by 2.
Worked Example 8 | Solve sin 2x=1/2
Within 0° to 720°, 2x=30°,150°,390°,510°. Therefore x=15°,75°,195°,255°.
Quadratic trigonometric equations
2sin²x−3sinx+1=0 is quadratic in sin x. Let u=sin x, factorise, then solve each trigonometric branch.
Worked Example 9 | Quadratic in sin x
(2sinx−1)(sinx−1)=0. Thus sinx=1/2 or 1. On 0°≤x≤360°, x=30°,90°,150°.
Reduce mixed-function equations
If an equation contains sin²x and cos²x, use sin²x+cos²x=1 to convert to one function. If tangent appears with sine or cosine, rewriting tan x may create a common structure.
Reduce the number of different trigonometric objects before solving.
Worked Example 10 | Convert to one function
sin²x=cos²x implies tan²x=1 where cosx≠0. Thus tanx=±1 and x=45°,135°,225°,315° on 0°≤x≤360°.
General solution versus interval solution
tanx=1 has general solution x=45°+180°n, where n is an integer. The function period controls the repeating family.
Inverse trigonometric functions give principal values
sin⁻¹(1/2)=30° does not mean 30° is the only angle with sine 1/2. The inverse function returns a chosen principal value. Periodic and quadrant reasoning completes an equation solution set.
Periodic modelling
Sine and cosine can model repeating systems such as rotation and oscillation. Amplitude describes the size of variation, period gives one full cycle, vertical shift gives the central level, and phase shift describes timing relative to a reference.
Worked Example 11 | Read a periodic model
h=5+2sin(30t°) has midline 5, amplitude 2 and period 360°/30°=12 time units. Maximum 7, minimum 3.
Trigonometry and calculus connect
Once sine and cosine are treated as functions, they can be differentiated and integrated. d/dx(sin x)=cos x, while ∫cosx dx=sinx+C. Composite forms such as sin3x require chain and reverse-chain reasoning.
Common trigonometry failure modes
| Error | Likely cause | Repair |
|---|---|---|
| Only one angle returned | Principal value mistaken for full set | Use quadrant and interval reasoning |
| Wrong numerical result | Degree/radian mismatch | State angle unit first |
| sin²x read as sin(x²) | Notation insecure | Rewrite as (sinx)² during learning |
| Identity proof moves both sides randomly | Proof treated as equation | Transform one side deliberately |
| Tangent uses 360° period | Function period not secure | Use 180° or π |
| sin2x interval not doubled | Inside angle ignored | Transform search interval first |
| Exact value converted too early | Calculator dependence | Preserve radicals and fractions |
Trigonometric equation decision tree
- Degrees or radians?
- What interval?
- Can the equation be reduced to one trig function?
- Is it linear or quadratic in the function value?
- Is the angle transformed?
- What is the reference angle?
- Which quadrants or periods are valid?
- Have all solutions been checked?
Examination control
Students often complete the algebra correctly and then lose marks by giving an incomplete set of angles, using the wrong interval, missing a repeated solution or rounding exact values too early. The final step should audit interval, unit, period, quadrants, transformed angle, exclusions and answer form.
In trigonometry, solving the equation is only half the task. Completing the solution set is the other half.
Independent practice
- Convert 225° to radians.
- Find exact sin150°.
- For y=4cos(2x)+3, state amplitude, period, maximum and minimum.
- Show that (1−sin²x)/cosx=cosx where defined.
- Solve sinx=−1/2 for 0°≤x≤360°.
- Solve cosx=1/2 for 0°≤x≤360°.
- Solve tanx=−1 for 0°≤x≤360°.
- Solve cos2x=0 for 0°≤x≤360°.
- Solve 2cos²x−cosx−1=0.
- Solve sin²x=3/4.
- For h=8+3sin(45t°), state amplitude, midline and period.
- Differentiate y=5sin(2x).
Explained answers
1. 5π/4. 2. 1/2. 3. amplitude 4, period 180°, maximum 7, minimum −1. 4. cos²x/cosx=cosx. 5. 210°,330°. 6. 60°,300°. 7. 135°,315°. 8. 45°,135°,225°,315°. 9. 0°,120°,240°,360°. 10. 60°,120°,240°,300°. 11. amplitude 3, midline 8, period 8. 12. 10cos(2x).
Teaching sequence
Begin with the unit circle, standard angles, reference angles and quadrant signs. Then build sine, cosine and tangent as graphs. Add transformed graphs, identities and equations only after the functions are visible. Identity work should develop representation choice; equation work should make interval control explicit. Finally mix trigonometry with calculus, functions and algebra so the chapter title can disappear without the structure disappearing.
How this guide connects to Secondary 4 A-Math
- Differentiation, tangents, normals and stationary points
- Integration, definite integrals and area under curves
- Exponential and logarithmic functions and equations
Final thought
Know the function. Read the interval. Preserve the identity. Return every valid angle.
Return to the Additional Mathematics Learning Hub.
