## Functions of a Right Triangle

From the right triangle shown below, the trigonometric functions of angle θ are defined as follows:

The Six Trigonometric Functions | |
---|---|

$\sin \theta = \dfrac{\text{opposite side}}{\text{hypotenuse}}$
$\sin \theta = \dfrac{a}{c}$ |
$\csc \theta = \dfrac{\text{hypotenuse}}{\text{opposite side}}$
$\csc \theta = \dfrac{c}{a}$ |

$\cos \theta = \dfrac{\text{adjacent side}}{\text{hypotenuse}}$
$\cos \theta = \dfrac{b}{c}$ |
$\sec \theta = \dfrac{\text{hypotenuse}}{\text{adjacent side}}$
$\sec \theta = \dfrac{c}{b}$ |

$\tan \theta = \dfrac{\text{opposite side}}{\text{adjacent side}}$
$\tan \theta = \dfrac{a}{b}$ |
$\cot \theta = \dfrac{\text{adjacent side}}{\text{opposite side}}$
$\cot \theta = \dfrac{b}{a}$ |

The above relationships can be written into acronym soh-cah-toa-cho-sha-cao.

**soh**=**s**ine of theta is equal to**o**pposite side over the**h**ypotenuse.**cah**=**c**osine of theta is equal to**a**djacent side over the**h**ypotenuse.**toa**=**t**angent of theta is equal to**o**pposite side over the**a**djacent side.**cho**=**c**osecant of theta is equal to**h**ypotenuse over the**o**pposite side.**sha**=**s**ecant of theta is equal to**h**ypotenuse over the**a**djacent side.**cao**=**c**otangent of theta is equal to**a**djacent side over the**o**pposite side.

## The Right Triangle

A triangle is said to be *right triangle* if one of its vertex-angle is equal to 90°. The 90° angle is called *right angle*. The figure below is a right triangle whose right angle is represented by a small square.

## The Pythagorean Theorem

In any right triangle, the sum of the squares of the two perpendicular sides is equal to the square of the longest side. The longest side is called *hypotenuse* of the right triangle.

For the right triangle above, the perpendicular legs are *a* and *b* and the hypotenuse is *c*. Hence,

## Derivation of Pythagorean Theorem

By Pythagoras • By Bhaskara • By U.S. Pres. James Garfield

## Click here to show or hide the derivation of Pythagorean Theorem

**Proved by Pythagoras**

Area of the large square = Area of four triangles + Area of small square

$A_{total} = A_{four \, \, triangles} + A_{small \, \, square}$

$(a + b)^2 = 4 \, \left( \frac{1}{2} ab \right) + c^2$

$a^2 + 2ab + b^2 = 2ab + c^2$

$a^2 + b^2 = c^2$

**Proved by Bhaskara**

Bhaskara (1114 - 1185) was an Indian mathematician and astronomer.

Area of the large square = Area of four triangles + Area of inner (smaller) square

$A_{total} = A_{four \, \, triangles} + A_{small \, \, square}$

$c^2 = 4 \, \left( \frac{1}{2} ab \right) + (b - a)^2$

$c^2 = 2ab + (b^2 - 2ab + a^2)$

$c^2 = 2ab + b^2 - 2ab + a^2$

$c^2 = b^2 + a^2$

**Proved by U.S. Pres. James Garfield**

Area of trapezoid = Area of 3 triangles

$\frac{1}{2}(a + b)(a + b) = \frac{1}{2}ab + \frac{1}{2}c^2 + \frac{1}{2}ab$

$(a + b)^2 = ab + c^2 + ab$

$a^2 + 2ab + b^2 = 2ab + c^2$

$a^2 + b^2 = c^2$

## The Oblique Triangle

A triangle is said to be an *oblique triangle* if none of its vertex-angle is equal to 90°.

The solution of oblique triangle is the solution of all triangles which includes the right triangle.

## The Sine law

In any triangle, the ratio of one side to the sine of its opposite angle is constant. This constant ratio is the diameter of the circle circumscribing the triangle.

For any triangles with vertex angles and corresponding opposite sides are *A*, *B*, *C* and *a*, *b*, *c*, respectively, the sine law is given by the formula...

## Click here to show or hide the derivation of Sine Law

_{B}as shown below. Expressing h

_{B}in terms of the side and the sine of the angle will lead to the formula of the sine law.

$\sin A = \dfrac{h_B}{c}$

$h_B = c \sin A$

$\sin C = \dfrac{h_B}{a}$

$h_B = a \sin C$

Equate the two h_{B}'s above:

$h_B = h_B$

$c \sin A = a \sin C$

$\dfrac{c}{\sin C} = \dfrac{a}{\sin A}$

To include angle B and side b in the above relationship, construct an altitude through C and label it h_{C} as shown below.

$\sin A = \dfrac{h_C}{b}$

$h_C = b \sin A$

$\sin B = \dfrac{h_C}{a}$

$h_C = a \sin B$

$h_C = h_C$

$b \sin A = a \sin B$

$\dfrac{b}{\sin B} = \dfrac{a}{\sin A}$

Thus,

Therefore, the ratio of one side to the sine of its opposite angle is constant.

**Note:**

The constant ratio above is the diameter of the circumscribing circle about the triangle.

## The Cosine Law

In any triangle, the square of one side is equal to the sum of the squares of the other two sides diminished by twice its product to the cosine of its included angle.

The following are the formulas for cosine law for any triangles with sides *a*, *b*, *c* and angles *A*, *B*, *C*, respectively.

$b^2 = a^2 + c^2 - 2ac\cos B$

$c^2 = a^2 + b^2 - 2ab\cos C$

## Click here to show or hide the derivation of Cosine Law

Cosine function for triangle ADB

$\cos A = \dfrac{x}{c}$

$x = c\cos A$

Pythagorean theorem for triangle ADB

$x^2 + h^2 = c^2$

$h^2 = c^2 - x^2$

Pythagorean theorem for triangle CDB

$(b - x)^2 + h^2 = a^2$

Substitute h^{2} = c^{2} - x^{2}

$(b - x)^2 + (c^2 - x^2) = a^2$

$(b^2 - 2bx + x^2) + (c^2 - x^2) = a^2$

$b^2 - 2bx + c^2 = a^2$

Substitute x = c cos A

$b^2 - 2b(c \cos A) + c^2 = a^2$

Rearrange:

The other two formulas can be derived in the same manner.