Trigonometric Identities

Equation of a unit circle

\(\(x^2+y^2=1\)\)

On a unit circle,
- the sin() of an angle is the x co-ordinate of the point
- the cos() of an angle is the y co-ordinate of the point
- the tan() of an angle is the gradient of the line drawn to the point from the origin

For any point that is not a multiple of 90

For \(0 < \theta < 90\), cos() is positive, sin() is positive, and tan is positive.
For \(90 < \theta < 180\), cos() is negetive, sin() is positive, and tan is negetive.
For \(180 < \theta < 270\), cos() is negetive, sin() is negetive, and tan is positive.
For \(270 < \theta < 360\), cos() is positive, sin() is negetive, and tan is positive.

Sine (red), Cosine (blue) and Tangent (green)

Tables of Trigonometric Values

	50	130	230	310
sin()	0.77	0.77	-0.77	-0.77
cos()	0.64	-0.64	-0.64	0.64
tan()	0.19	-0.19	0.19	-0.19

	30	150	210	330
sin()	\(1\over2\)	\(1\over2\)	\(- 1\over2\)	\(- 1\over2\)
cos()	\(\sqrt{3}\over2\)	\(- \sqrt{3}\over2\)	\(- \sqrt{3}\over2\)	\(\sqrt{3}\over2\)
tan()	\(\sqrt{3}\over3\)	\(- \sqrt{3}\over3\)	\(\sqrt{3}\over3\)	\(- \sqrt{3}\over3\)

More Trigonometric Identities

Info

First rule: \(\tan\theta\equiv \frac{\sin\theta}{\cos\theta}\)

Second rule: \(\sin^2\theta+\cos^{2}\theta\equiv 1\)

Third rule: \(a^2-b^2=(a+b)(a-b)\)

Fourth rule: \(a^4-b^4=(a^2+b^2)(a^2-b^2)\)

Examples

Prove that \(\frac{\tan x\cos x}{\sqrt{ 1-\cos^{2}x}} \equiv 1\)

Square both sides:
\(\frac{\tan^2x\cos^2x}{1-\cos^2x}\equiv1\)

Use the first rule:
\(\frac{\sin^2x\cos^2x}{\cos^2x+1-\cos^2x} \equiv 1\)

Cancel the \(\cos ^2x\) on the bottom:
\(\sin^2x\cos^2x \equiv 1\)

Use the second rule:
\(1 \equiv 1\)

Prove that \(\frac{\cos^4x-\sin^4x}{\cos^2x} \equiv 1-\tan^2x\)

Expand brackets:
\(\frac{(\cos^2x+\sin^2x)(\cos^2x-\sin^2x)}{\cos^2x} \equiv 1-\tan^2x\)

\(\sin^2x+\cos^2x \equiv 1 \therefore\) we can get rid of it:
\(\frac{\cos^2x-\sin^2x}{\cos^2x} \equiv \frac{\cos^2x}{\cos^2x}-\frac{\sin^2x}{\cos^2x}\)

\(\equiv 1-\tan^2x \equiv RHS\)