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Trigonometric identities and examples

trigonometric identities and examples

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)$:

pythagorean identity

Using this identity we can create two more. First, if you divide whole equation with $ cos^2(\alpha)$:

tangent and secant in the pythagorean identity

And the second, if you divide whole equation with $ sin^2(\alpha)$:

cotangent and cosecant in the pythagorean identity

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.

two adjacents angles

Now draw perpendicular line on side (AB) that goes through point D. Mark that intersection with E.

draw perpendicular line in adjacent

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.

draw perpendicular line on side ac

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.

intersection line

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.

final proof

On the other hand EA = GA – FH. Equivalently we get:

cos alpha plus beta

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.

law of sine

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$.

solve task using law of sine
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.

law of cosine

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

double angle identities formulas

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

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

sum identities trigonometry formula

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, 804 hits)

  Area of triangle (309.2 KiB, 487 hits)

  Area of regular polygon - Side known (289.9 KiB, 504 hits)

  Area of regular polygon - A perimeter available (307.5 KiB, 429 hits)

  Trigonometric equations (184.3 KiB, 697 hits)

  The Law of Sine (340.3 KiB, 470 hits)

  The Law of Cosine (326.4 KiB, 417 hits)

  Area of triangle (234.5 KiB, 395 hits)

  Angle sum and difference identities (98.4 KiB, 446 hits)

  Double-angle and half-angle identities (146.3 KiB, 522 hits)