Trigonometry: trig functions, identities, equations, solutions, graphs

Trigonometry – Functions, Identities, Equations, Solutions & Graphs (Cambridge AS & A‑Level 9709)

1. Fundamental Trigonometric Functions

For an angle θ (measured in degrees or radians) the six trigonometric functions are defined using a right‑angled triangle or, more generally, the unit circle.

Function	Right‑angled triangle definition	Unit‑circle definition
sin θ	opposite ⁄ hypotenuse	y‑coordinate of the point (cos θ, sin θ)
cos θ	adjacent ⁄ hypotenuse	x‑coordinate of the point (cos θ, sin θ)
tan θ	opposite ⁄ adjacent	sin θ ⁄ cos θ
csc θ	hypotenuse ⁄ opposite	1 ⁄ sin θ
sec θ	hypotenuse ⁄ adjacent	1 ⁄ cos θ
cot θ	adjacent ⁄ opposite	cos θ ⁄ sin θ

2. Exact Values for the Common Angles

θ	sin θ	cos θ	tan θ	csc θ	sec θ	cot θ
0° (0)	0	1	0	–	1	–
30° (π⁄6)	½	√3⁄2	1⁄√3	2	2⁄√3	√3
45° (π⁄4)	√2⁄2	√2⁄2	1	√2	√2	1
60° (π⁄3)	√3⁄2	½	√3	2⁄√3	2	1⁄√3
90° (π⁄2)	1	0	–	1	–	0

3. Inverse Trigonometric Functions (Principal Values)

• sin⁻¹ x (arcsin x)  Domain = [‑1, 1], Range = [‑π⁄2, π⁄2] (or [‑90°, 90°])
• cos⁻¹ x (arccos x)  Domain = [‑1, 1], Range = [0, π] (or [0°, 180°])
• tan⁻¹ x (arctan x)  Domain = ℝ,  Range = (‑π⁄2, π⁄2) (or (‑90°, 90°))

These principal‑value definitions are used when solving simple equations and when writing answers in the required interval.

4. Core Trigonometric Identities

4.1 Pythagorean Identities

\[ \sin^{2}\theta+\cos^{2}\theta=1,\qquad 1+\tan^{2}\theta=\sec^{2}\theta,\qquad 1+\cot^{2}\theta=\csc^{2}\theta \]

4.2 Reciprocal Identities

\[ \csc\theta=\frac{1}{\sin\theta},\quad \sec\theta=\frac{1}{\cos\theta},\quad \cot\theta=\frac{1}{\tan\theta} \]

4.3 Quotient Identities

\[ \tan\theta=\frac{\sin\theta}{\cos\theta},\qquad \cot\theta=\frac{\cos\theta}{\sin\theta} \]

4.4 Co‑function Identities (θ in radians)

\[ \begin{aligned} \sin\!\Bigl(\tfrac{\pi}{2}-\theta\Bigr)&=\cos\theta, & \cos\!\Bigl(\tfrac{\pi}{2}-\theta\Bigr)&=\sin\theta,\\[2mm] \tan\!\Bigl(\tfrac{\pi}{2}-\theta\Bigr)&=\cot\theta, & \cot\!\Bigl(\tfrac{\pi}{2}-\theta\Bigr)&=\tan\theta,\\[2mm] \sec\!\Bigl(\tfrac{\pi}{2}-\theta\Bigr)&=\csc\theta, & \csc\!\Bigl(\tfrac{\pi}{2}-\theta\Bigr)&=\sec\theta. \end{aligned} \]

4.5 Even–Odd (Parity) Identities

\[ \begin{aligned} \sin(-\theta)&=-\sin\theta, & \cos(-\theta)&=\cos\theta,\\ \tan(-\theta)&=-\tan\theta, & \sec(-\theta)&=\sec\theta,\\ \csc(-\theta)&=-\csc\theta, & \cot(-\theta)&=-\cot\theta. \end{aligned} \]

5. Compound‑Angle, Double‑Angle & Half‑Angle Identities

5.1 Sum & Difference (Compound‑Angle) Identities

\[ \begin{aligned} \sin(\alpha\pm\beta)&=\sin\alpha\cos\beta\pm\cos\alpha\sin\beta,\\[2mm] \cos(\alpha\pm\beta)&=\cos\alpha\cos\beta\mp\sin\alpha\sin\beta,\\[2mm] \tan(\alpha\pm\beta)&=\frac{\tan\alpha\pm\tan\beta}{1\mp\tan\alpha\tan\beta}. \end{aligned} \]

5.2 Double‑Angle Identities

\[ \begin{aligned} \sin2\theta &= 2\sin\theta\cos\theta,\\[2mm] \cos2\theta &= \cos^{2}\theta-\sin^{2}\theta = 2\cos^{2}\theta-1 = 1-2\sin^{2}\theta,\\[2mm] \tan2\theta &= \frac{2\tan\theta}{1-\tan^{2}\theta}. \end{aligned} \]

5.3 Half‑Angle Identities (derived from the double‑angle formulas)

\[ \begin{aligned} \sin^{2}\frac{\theta}{2}&=\frac{1-\cos\theta}{2}, & \cos^{2}\frac{\theta}{2}&=\frac{1+\cos\theta}{2},\\[2mm] \tan\frac{\theta}{2}&=\frac{\sin\theta}{1+\cos\theta} =\frac{1-\cos\theta}{\sin\theta}. \end{aligned} \]

6. Solving Simple Trigonometric Equations

The Cambridge syllabus expects students to solve equations without writing the full general solution (i.e. only the solutions required in the given interval).

6.1 General Procedure (for \(0\le\theta<2\pi\) or \(0^\circ\le\theta<360^\circ\))

1. Rewrite. Use identities to express the equation in terms of a single trig function.
2. Isolate the function. Bring the equation to the form \(f(\theta)=k\).
3. Check the range. Verify that \(k\) lies in the range of \(f\) (e.g. \(|k|\le1\) for sin θ or cos θ).
4. Find the principal angles. Use the inverse functions (sin⁻¹, cos⁻¹, tan⁻¹) to obtain the first solution(s) in the required interval.
5. Use symmetry. Apply the appropriate symmetry of the function to obtain any additional solutions in the interval.
6. Check. Substitute each answer back into the original equation to discard extraneous values.

6.2 Worked Example 1 – A Quadratic in sin θ

Solve \(\;2\sin^{2}\theta-\sqrt{3}\sin\theta-1=0\;\) for \(0\le\theta<2\pi\).

1. Let \(x=\sin\theta\). The equation becomes \(2x^{2}-\sqrt{3}x-1=0\).
2. Quadratic formula: \(\displaystyle x=\frac{\sqrt{3}\pm\sqrt{(\sqrt{3})^{2}+8}}{4} =\frac{\sqrt{3}\pm\sqrt{11}}{4}\).
3. Numerical values: \(x_{1}\approx0.933\) (allowed), \(x_{2}\approx-0.683\) (allowed).
4. Principal angles (using arcsin):
   • \(\theta_{1}= \sin^{-1}(0.933)\approx1.21\text{ rad}\) (≈ 69°). Second solution in the interval: \(\pi-\theta_{1}\approx1.93\text{ rad}\) (≈ 111°).
   • \(\theta_{2}= \sin^{-1}(-0.683)\approx-0.75\text{ rad}\) → add \(2\pi\) → \(5.53\text{ rad}\) (≈ 317°). Second solution: \(\pi-\theta_{2}\approx3.89\text{ rad}\) (≈ 223°).
5. Solutions in \([0,2\pi)\): \[ \theta\approx1.21,\;1.93,\;3.89,\;5.53\text{ rad} \quad\bigl(\text{or }69^\circ,111^\circ,223^\circ,317^\circ\bigr). \]

6.3 Worked Example 2 – Using a Double‑Angle Identity

Solve \(\cos2\theta = \frac12\) for \(0^\circ\le\theta<360^\circ\).

1. Write the double‑angle as a single‑angle expression: \(\cos2\theta = 2\cos^{2}\theta-1\).
2. Set up the equation: \(2\cos^{2}\theta-1=\frac12\) → \(\cos^{2}\theta=\frac34\).
3. Take square roots: \(\cos\theta = \pm\frac{\sqrt3}{2}\).
4. Principal angles for \(\cos\theta = \frac{\sqrt3}{2}\) are \(30^\circ\) and \(330^\circ\). For \(\cos\theta = -\frac{\sqrt3}{2}\) they are \(150^\circ\) and \(210^\circ\).
5. All four angles lie in the required interval, so \[ \theta = 30^\circ,\;150^\circ,\;210^\circ,\;330^\circ. \]

7. Graphs of the Basic Trigonometric Functions

7.1 Standard Forms

• \(y = a\sin(bx + c) + d\)
• \(y = a\cos(bx + c) + d\)
• \(y = a\tan(bx + c) + d\)

7.2 Key Characteristics (for \(y = a\sin(bx + c) + d\) and \(y = a\cos(bx + c) + d\))

Feature	Formula	Effect on the graph
Amplitude	\(|a|\)	Vertical stretch/compression; reflection in the x‑axis if \(a<0\).
Period	\(\displaystyle \frac{2\pi}{|b|}\)	Horizontal compression (|b|>1) or stretch (|b|<1).
Phase shift	\(-\dfrac{c}{b}\)	Shift right if the quantity is positive, left if negative.
Vertical shift	\(d\)	Moves the whole curve up (positive d) or down (negative d).

7.3 Characteristics of the Tangent Graph

Feature	Formula / Description
Period	\(\displaystyle \frac{\pi}{|b|}\)
Amplitude	Not defined (graph is unbounded).
Phase shift	\(-\dfrac{c}{b}\) (same rule as sine & cosine).
Vertical shift	\(d\) (moves the mid‑line up or down).
Asymptotes	Where \(\cos(bx+c)=0\) → \(x = \frac{\pi}{2b}-\frac{c}{b}+k\frac{\pi}{b}\), \(k\in\mathbb Z\).

7.4 Example – Transformations

1. Function: \(y = 2\sin\bigl(3x-\tfrac{\pi}{4}\bigr)+1\)
   • Amplitude = 2
   • Period = \(\dfrac{2\pi}{3}\)
   • Phase shift = \(\displaystyle \frac{\pi}{12}\) to the right (because \(-c/b = \pi/12\))
   • Vertical shift = +1
2. Function: \(y = -\dfrac12\sec\!\left(\tfrac{x}{2}+\tfrac{\pi}{6}\right)\)
   • Amplitude (stretch) = \(\frac12\) (compresses towards the x‑axis)
   • Reflection in the x‑axis (negative sign)
   • Period = \(\dfrac{2\pi}{\tfrac12}=4\pi\)
   • Phase shift = \(-\tfrac{\pi}{3}\) (left)

8. Quick Reference – Summary Tables

8.1 Core Identities at a Glance

\[ \begin{aligned} \sin(\alpha\pm\beta)&=\sin\alpha\cos\beta\pm\cos\alpha\sin\beta,\\ \cos(\alpha\pm\beta)&=\cos\alpha\cos\beta\mp\sin\alpha\sin\beta,\\ \tan(\alpha\pm\beta)&=\frac{\tan\alpha\pm\tan\beta}{1\mp\tan\alpha\tan\beta},\\[4pt] \sin2\theta&=2\sin\theta\cos\theta,\\ \cos2\theta&=\cos^{2}\theta-\sin^{2}\theta =2\cos^{2}\theta-1 =1-2\sin^{2}\theta,\\ \tan2\theta&=\frac{2\tan\theta}{1-\tan^{2}\theta},\\[4pt] \sin^{2}\frac{\theta}{2}&=\frac{1-\cos\theta}{2},\qquad \cos^{2}\frac{\theta}{2}=\frac{1+\cos\theta}{2},\\ \tan\frac{\theta}{2}&=\frac{\sin\theta}{1+\cos\theta} =\frac{1-\cos\theta}{\sin\theta}. \end{aligned} \]

8.2 Periodicity & Symmetry

• \(\sin(\theta+2\pi)=\sin\theta\), \(\cos(\theta+2\pi)=\cos\theta\)
• \(\tan(\theta+\pi)=\tan\theta\), \(\cot(\theta+\pi)=\cot\theta\)
• Even functions: \(\cos(-\theta)=\cos\theta,\; \sec(-\theta)=\sec\theta\)
• Odd functions: \(\sin(-\theta)=-\sin\theta,\; \tan(-\theta)=-\tan\theta\) (and the corresponding reciprocals).

8.3 Common Exact Values (re‑listed for quick lookup)

θ	sin θ	cos θ	tan θ
0° (0)	0	1	0
30° (π⁄6)	½	√3⁄2	1⁄√3
45° (π⁄4)	√2⁄2	√2⁄2	1
60° (π⁄3)	√3⁄2	½	√3
90° (π⁄2)	1	0	–

9. Summary of What You Must Be Able to Do (Cambridge AS & A‑Level)

• Sketch accurate graphs of sin, cos and tan for any angle (degrees or radians) and identify amplitude, period, phase and vertical shift.
• State and use the exact values for 0°, 30°, 45°, 60°, 90° (and their radian equivalents).
• Write and evaluate principal‑value inverse functions sin⁻¹, cos⁻¹, tan⁻¹.
• Recall and apply all core identities listed in sections 4 and 5.
• Solve simple trigonometric equations in the required interval, using identities and inverse functions, and check each answer.
• Transform the basic sine, cosine and tangent graphs using the parameters a, b, c and d.