Trigonometric Ratios & Identities
The geometry of the circle turned into algebra — six ratios and the web of identities that connect them
- To measure angles in radians and move freely between radians and degrees.
- The six ratios on the unit circle, their signs by quadrant (ASTC), and standard values.
- The three Pythagorean identities and the allied-angle reduction rules.
- The sum, difference, multiple and sub-multiple angle formulae.
- Product-to-sum and sum-to-product transformations.
- The range of \(a\sin\theta+b\cos\theta\) — the key to max/min problems.
Angles & Radians
Degrees are a human convention; radians are the angle measure mathematics actually prefers, because they make arc length and calculus formulae clean. One radian is the angle subtended at the centre of a circle by an arc equal in length to the radius.
So \(1^{\circ}=\dfrac{\pi}{180}\) rad and \(1\text{ rad}=\dfrac{180}{\pi}\approx57.3^{\circ}\). The arc-length formula \(s=r\theta\) holds only when \(\theta\) is in radians — one of the main reasons radians are the default.
The Six Ratios on the Unit Circle
Place an angle \(\theta\) at the centre of a unit circle, measured anticlockwise from the positive \(x\)-axis. The point where its arm meets the circle is \(P(\cos\theta,\sin\theta)\) — that single point defines every ratio.
| Ratio | Reciprocal |
|---|---|
| \(\sin\theta\) (\(y\)) | \(\csc\theta=\dfrac{1}{\sin\theta}\) |
| \(\cos\theta\) (\(x\)) | \(\sec\theta=\dfrac{1}{\cos\theta}\) |
| \(\tan\theta=\dfrac{\sin\theta}{\cos\theta}\) | \(\cot\theta=\dfrac{\cos\theta}{\sin\theta}\) |
Signs in the Four Quadrants
Because sine is the \(y\)-coordinate and cosine the \(x\)-coordinate, each ratio's sign follows the sign of that coordinate in the quadrant.
In quadrant I every ratio is positive; thereafter only the named one (and its reciprocal) stays positive. A mnemonic: "Add Sugar To Coffee."
The Fundamental Identities
From \(x^2+y^2=1\) on the unit circle, the whole Pythagorean family follows at once.
Allied Angles & Reduction
Angles like \(-\theta,\ 90^{\circ}\pm\theta,\ 180^{\circ}\pm\theta\) are "allied" to \(\theta\). Two simple rules collapse all of them.
For \(90^{\circ}\pm\theta\) and \(270^{\circ}\pm\theta\), the ratio changes to its co-ratio (sin↔cos, tan↔cot, sec↔csc). For \(180^{\circ}\pm\theta\) and \(360^{\circ}\pm\theta\), it stays the same. The sign is whatever the original function has in the quadrant the new angle lands in. Also: \(\sin(-\theta)=-\sin\theta\) (odd), \(\cos(-\theta)=\cos\theta\) (even).
Sum & Difference Formulae
These are the heart of the chapter — every later formula descends from them.
In \(\cos(A+B)\) the cross-term is minus: \(\cos A\cos B-\sin A\sin B\). Sine keeps the same sign as the angle; cosine reverses it. Write the formula out fully before substituting and this never bites.
Multiple-Angle Formulae
Put \(B=A\) in the sum formulae and the double-angle identities appear; push once more for the triple-angle versions.
| Angle | Identity |
|---|---|
| \(\sin2A\) | \(2\sin A\cos A=\dfrac{2\tan A}{1+\tan^{2}A}\) |
| \(\cos2A\) | \(\cos^{2}A-\sin^{2}A=2\cos^{2}A-1=1-2\sin^{2}A\) |
| \(\tan2A\) | \(\dfrac{2\tan A}{1-\tan^{2}A}\) |
| \(\sin3A\) | \(3\sin A-4\sin^{3}A\) |
| \(\cos3A\) | \(4\cos^{3}A-3\cos A\) |
Sub-multiple Angles
Reading the double-angle identities backwards expresses ratios of \(\theta\) through \(\tfrac{\theta}{2}\) — invaluable for integration and for the "\(t\)-substitution."
With \(t=\tan\tfrac{\theta}{2}\), every ratio becomes a rational function of \(t\). This single substitution turns many trigonometric equations and integrals into algebra.
Product & Sum Transformations
Adding and subtracting the sum formulae converts products into sums; reversing converts sums into products. Both directions are constantly useful — products are easy to integrate, sums easy to factor.
| Product → Sum | Sum → Product |
|---|---|
| \(2\sin A\cos B=\sin(A{+}B)+\sin(A{-}B)\) | \(\sin C+\sin D=2\sin\tfrac{C+D}{2}\cos\tfrac{C-D}{2}\) |
| \(2\cos A\cos B=\cos(A{+}B)+\cos(A{-}B)\) | \(\cos C+\cos D=2\cos\tfrac{C+D}{2}\cos\tfrac{C-D}{2}\) |
| \(2\sin A\sin B=\cos(A{-}B)-\cos(A{+}B)\) | \(\cos C-\cos D=-2\sin\tfrac{C+D}{2}\sin\tfrac{C-D}{2}\) |
Strategies & Standard Results
A few reliable moves and one result that settles most maxima–minima questions.
When an identity tangles, rewrite every ratio as sine and cosine and simplify the resulting fraction.
Pair terms with equal "weight" so the sum-to-product formulae produce a common factor.
In a triangle, conditional identities like \(\tan A+\tan B+\tan C=\tan A\tan B\tan C\) shortcut whole problems.
Write \(a\sin\theta+b\cos\theta=R\sin(\theta+\varphi)\) with \(R=\sqrt{a^2+b^2}\); the extreme values are \(\pm R\). This one fact answers nearly every trigonometric max/min question.
Putting It to Work
Problem. Find \(\cos15^{\circ}\).
Solution. Write \(15^{\circ}=45^{\circ}-30^{\circ}\) and use the difference formula:
Problem. Prove \(\dfrac{\sin3A}{\sin A}-\dfrac{\cos3A}{\cos A}=2\).
Solution. Combine over a common denominator; the numerator is \(\sin3A\cos A-\cos3A\sin A=\sin(3A-A)=\sin2A\):
Problem. If \(\sin A=\tfrac35\) and \(A\) is acute, find \(\sin2A\) and \(\cos2A\).
Solution. \(\cos A=\tfrac45\). Then
Problem. Simplify \(\dfrac{\sin5A+\sin3A}{\cos5A+\cos3A}\).
Solution. Both numerator and denominator factor with the same average angle \(4A\):
Problem. Find the maximum and minimum of \(3\sin\theta+4\cos\theta\).
Solution. Here \(R=\sqrt{3^2+4^2}=5\), so the expression equals \(5\sin(\theta+\varphi)\):
Problem. If \(A+B+C=\pi\), prove \(\tan A+\tan B+\tan C=\tan A\tan B\tan C\).
Solution. Since \(A+B=\pi-C\), take tangents: \(\tan(A+B)=-\tan C\).
Chapter Summary
\(\pi=180^{\circ}\); \(s=r\theta\) needs radians. \(P=(\cos\theta,\sin\theta)\) on the unit circle.
ASTC: All, Sin, Tan, Cos positive in quadrants I–IV.
\(\sin^2+\cos^2=1\), \(1+\tan^2=\sec^2\), \(1+\cot^2=\csc^2\).
\(\sin(A\pm B),\cos(A\pm B),\tan(A\pm B)\) — mind the sign rules.
Double and triple angles; \(\cos2A\) in three forms; the \(t=\tan\tfrac{\theta}{2}\) substitution.
Product↔sum formulae; \(a\sin\theta+b\cos\theta\in[-\sqrt{a^2+b^2},\sqrt{a^2+b^2}]\).
Problems
Reduce to sine and cosine when stuck. Difficulty rises down the list.
- Convert \(150^{\circ}\) to radians and \(\tfrac{3\pi}{4}\) to degrees.
- Find the exact value of \(\sin75^{\circ}\) and \(\tan15^{\circ}\).
- If \(\cos\theta=-\tfrac{5}{13}\) and \(\theta\) is in quadrant II, find the other five ratios.
- Prove \(\dfrac{1+\sin2A}{\cos2A}=\dfrac{\cos A+\sin A}{\cos A-\sin A}\).
- Express \(\sin5\theta+\sin\theta\) as a product and simplify \(\dfrac{\sin5\theta-\sin\theta}{\cos5\theta+\cos\theta}\).
- Show that \(\cos20^{\circ}\cos40^{\circ}\cos80^{\circ}=\tfrac18\).
- Prove \(\sin3A=3\sin A-4\sin^{3}A\) from the sum formula.
- Find the maximum value of \(5\sin\theta-12\cos\theta+7\).
- If \(\tan\tfrac{\theta}{2}=t\), express \(\dfrac{1-\cos\theta+\sin\theta}{1+\cos\theta+\sin\theta}\) in terms of \(t\) and simplify.
- In a triangle, prove \(\sin2A+\sin2B+\sin2C=4\sin A\sin B\sin C\).
- Prove that \(\cos^{2}A+\cos^{2}(A+120^{\circ})+\cos^{2}(A-120^{\circ})=\tfrac32\).
- Find the range of \(f(\theta)=\sin^{4}\theta+\cos^{4}\theta\).