Pythagorean identity
The main Pythagorean identity is the notation of Pythagorean Theorem in made in terms of unit circle, and a specific angle. Let’s draw unit circle, set some angle arbitrarily and mark the right triangle determined by those informations.
Now we got a right triangle with legs, whose lengths are $ sin(\alpha)$ and $ cos(\alpha)$, and hypotenuse whose length is equal to 1. Since this triangle is right, we can use Pythagorean Theorem which leads us to:
- Using this identity we can create two more. First, if you divide whole equation with $ cos^2(\alpha)$:
Using this identity we can create two more. First, if you divide whole equation with $ cos^2(\alpha)$:
And the second, if you divide whole equation with $ sin^2(\alpha)$:
Example 1. If $ cot \alpha = 2$ find $ csc^2(\alpha)$
$ 1 + cot^2(\alpha) = csc^2(\alpha)$
$ 1 + 2 = csc^2(\alpha)$
$ csc^2(\alpha) = 3$
Example 2. If $ sin(x) = 0.25$ find $ cos(x)$.
$ sin^2(x) + cos^2(x) = 10.25^2 + cos^2(x) = 1$
$ cos^2(x) = 1 – 0.0625$
$ cos^2(x) = 0.9375 \rightarrow cos(x_1) = 0.968$, $ cos(x_2) = – 0.968$
Example 3. If $ sin(x) = 0.5$ and $ cos(x) = 0.2$ find $ sec^2(x)$.
$ tan^2(x) + 1 = sec^2(x)$
$\frac{sin^2(x)}{cos^2(x)} + 1 = sec^2(x)$
$ sec^2(x) = 1.29$
Angle addition formulas
Addition formula of difference for cosine
For every two real numbers it is valid that: $ cos (x – y) = cos(x)cos(y) + sin(x)sin(y)$
Example. If you get an angle that you can write as a difference of two angles whose trigonometric values you know, using this formula, you can calculate its value without calculator.
If you have to calculate $ cos(15^{\circ})$ you can write that as $ cos(45^{\circ} – 30^{\circ})$ and calculate the rest by the formula.
$cos (45^{\circ} – 30^{\circ}) = cos(45^{\circ})cos(30^{\circ}) + sin(30^{\circ})cos(45^{\circ}) = \frac{\sqrt{2}}{2} * \frac{\sqrt{3}}{2} + \frac{1}{2} * \frac{\sqrt{2}}{2} = \frac{\sqrt{6} + \sqrt{2}}{4}$
Addition formula of the sum for cosine
For every two real numbers it is valid that: $cos(x + y) = cos(x)cos(y) – sin(x)sin(y)$
Example. Find $ cos(75^{\circ})$.
$ cos 75^{\circ} = cos(45^{\circ} + 30^{\circ}) = cos(45^{\circ})cos(30^{\circ}) – sin(45^{\circ})sin(30^{\circ}) = \frac{\sqrt{6} – \sqrt{2}}{4}$
Addition formula for sine of difference
For every two real numbers it is valid that: $ sin(x – y) = sin(x)cos(y) – cos(x)sin(y)$
Example:
$sin (\frac{\pi}{2} – x) = sin(\frac{\pi}{2})cos(x) – cos(\frac{\pi}{2})sin(x) = 1cos(x) – 0sin(x) = cos(x)$
Addition formula for sine of the sum
For every two real numbers it is valid that: $sin(x + y) = sin(x)cos(y) + cos(x)sin(y)$
Example 1.: $sin 120^{\circ} = sin(90^{\circ} + 30^{\circ}) = sin(90^{\circ})cos(30^{\circ}) + cos(90^{\circ})sin(30^{\circ}) = \frac{\sqrt{3}}{2}$
There are also trigonometric functions of tangent and cotangent but they can be extracted from the sine and cosine.
$tan(x + y) = \frac{tan(x) + tan(y)}{1 – tan(x)tan(y)}$
$tan(x – y) = \frac{tan(x) + tan(y)}{1 + tan(x)tan(y)}$
Example 2.: Using these theorems prove following
$ sin(x + y) + sin(x – y) = 2 sin(x)cos(y)$
$ sin(x + y) + sin(x – y) = sin(x)cos(y) + cos(x)sin(y) + sin(x)cos(y) – cos(x)sin(y) = sin(x)cos(y) + sin(x)cos(y) = 2sin(x)cos(y)$
$ cos(x + y)cos(x – y) = cos^2(x) – sin^2(y)$
$ [cos(x)cos(y) – sin(x)sin(y)][cos(x)cos(y) + sin(x)sin(y)] = cos^2(x)cos^2(y) – sin^2(x)sin^2(y) = cos^2(x)(1 – sin^2(y)) – sin^2(x)sin^2(y) = cos^2(x) – cos^2(x)sin^2(y) – sin^2(x)sin^2(y) = cos^2(x) – sin^2(y)(cos^2(x) + sin^2(x)) = cos^2(x) – sin^2(y)$
Angle addition formula proofs
First, draw two adjacent angles α and β.
Now draw perpendicular line on side (AB) that goes through point D. Mark that intersection with E.
Now we got a right triangle and we can conclude that $ sin(\alpha + \beta) = \frac{\mid DE \mid}{DA}$ | (opposite over hypotenuse).
Next thing you should do is draw perpendicular line on the side (AC) that goes through point D. Mark that intersection with F.
The last thing you need to do is to draw perpendicular line to the side AB through point F. Mark that intersection with G and another perpendicular line to the side DE through point F.
Let’s now observe angles GAF and FDE. By the Angles Subtended by Same Arc theorem these angles are equal. And also, the straight line AC crosses parallel lines HF and AB, Angle HFA is also equal to α.
On the other hand EA = GA – FH. Equivalently we get:
If you’d want to prove the difference formula, you’d simply replace + $\beta$ with + (- $\beta$) using following identities:
$ cos(-\beta) = cos(\beta)$
$ sin(-\beta) = – sin(\beta)$
Law of cosines and law of sines
The Law of Sines
Ratios of angles and their opposite sides in triangle are equal.
This law can be applied to any sort of triangles, and is very useful when quick and precise solution is needed.
Example: In a triangle ABC, find $ b$ if $ \alpha = 30^{\circ}$, $ \beta = 60^{\circ}$, $ a = 5$.
The Law of Cosines
The law of cosines is used when you have either three sides and looking for an angle, or have two sides and an angle and looking for the third side.
Note that on the right side of the equation always stands the angle opposite to the side on the left.
Example 1. Find α in a triangle ABC if $ a = 5, b = 4, c = 3$.
A is opposite to α:
$ a^2 = b^2 + c^2 – 2bc \cdot cos(\alpha)$
$ 25 = 16 + 9 – 24 cos(\alpha)$
$ cos(\alpha) = 0 \rightarrow \alpha = \frac{\pi}{2} + k\pi$
Trigonometric identities
Double – angle identities
Example 1. Prove that $ sin (90^{\circ}) = 1$ using double angle identities.
$sin(90^{\circ}) = sin(2 * 45^{\circ}) = 2 sin(45^{\circ})cos(45^{\circ}) = 2 * \frac{\sqrt{2}}{2} * \frac{\sqrt{2}}{2} = 2 * \frac{1}{2} = 1$
Example 2. Calculate $cos(120^{\circ})$.
$cos(120^{\circ}) = cos(2 * 60^{\circ}) = cos^2(60^{\circ}) – sin^2(60^{\circ})= \frac{1}{4} – \frac{3}{4} = – \frac{1}{2}$
Half – angle identities
Example 1. Calculate $cos(15^{\circ})$
$cos(15^{\circ}) = \frac{cos(30^{\circ})}{2} = \pm \sqrt{\frac{1 – cos(30^{\circ})}{2}} = \pm 0.259$
Example 2. Calculate $tan(22,5^{\circ})$
$tan(22,5^{\circ}) = tan(\frac{45^{\circ}}{2}) = \pm \sqrt{\frac{(1 – cos(45^{\circ})}{(1 + cos(45^{\circ})}} = \pm ± 0.414$
Product identities
There are more formulas that can help you to simplify complex trigonometric terms.
We have our well known addition formulas:
$ sin(x + y) = sin(x)cos(y) + cos(x)sin(y)$
$ sin(x – y) = sin(x)cos(y) – cos(x)sin(y)$
If we add those two equations and divide it by two, we’ll get
$sin(x)cos(y) = \frac{1}{2}(sin(x + y) + sin(x – y))$
$cos(x)sin(y) = \frac{1}{2}(sin(x + y) – sin(x – y))$
Similarly we get
$cos(x)cos(y) = \frac{1}{2}(cos(x – y) + cos(x + y))$
$sin(x)sin(y) = \frac{1}{2}(cos(x – y) – cos(x + y))$
Example 1. Calculate $sin(60^{\circ})cos(30^{\circ})$ using the product identities.
$sin(60^{\circ})cos(30^{\circ}) = \frac{1}{2}(sin(90^{\circ}) + sin(30^{\circ})) = \frac{1}{2} * \frac{3}{2} = \frac{3}{4}$
Example 2. Calculate $sin(60^{\circ})sin(30^{\circ})$ using the product identities.
$sin(60^{\circ})sin(30^{\circ}) = \frac{1}{2}(cos(30^{\circ}) – cos(90^{\circ})) = \frac{\sqrt{3}}{4}$
Sum identities
Example 1. Simplify $sin(30^{\circ} + x)cos(30^{\circ} – x)$.
This expression looks a lot like our first sum identity, but not quite. We’ll simply make it look like one and adjust so everything fits.
$sin(\frac{60^{\circ} + 2x}{2})cos(\frac{60^{\circ} + 2x}{2}) = \frac{1}{2}(sin(60^{\circ}) + sin(2x)$
Example 2. Expand using sum identities $cos(x + 20^{\circ}) – cos(60^{\circ})$
$cos(x + 20^{\circ}) – cos(60^{\circ}) = – 2 sin(\frac{x + 20^{\circ} + 60^{\circ}}{2})sin(\frac{x + 20^{\circ} – 60^{\circ}}{2}) = – 2 sin(\frac{x + 80^{\circ}}{2})sin(\frac{x – 40^{\circ}}{2})$
Example 3. Calculate using sum identities $sin(30^{\circ}) – sin(30^{\circ})$.
$sin(30^{\circ}) – sin(60^{\circ}) = 2 cos(\frac{90^{\circ}}{2})sin(\frac{- 30^{\circ}}{2}) = -2 cos(45^{\circ})sin(15^{\circ}) = – 0.366$
rigonometric identities and examples worksheets
Trigonometric ratios in a right triangles (171.3 KiB, 1,537 hits)
Area of triangle (309.2 KiB, 766 hits)
Area of regular polygon - Side known (289.9 KiB, 796 hits)
Area of regular polygon - A perimeter available (307.5 KiB, 732 hits)
Trigonometric equations (184.3 KiB, 1,179 hits)
The Law of Sine (340.3 KiB, 822 hits)
The Law of Cosine (326.4 KiB, 691 hits)
Area of triangle (234.5 KiB, 681 hits)
Angle sum and difference identities (98.4 KiB, 805 hits)
Double-angle and half-angle identities (146.3 KiB, 849 hits)
