Functions of a Right Triangle

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

Right triangle with sides a, b, and c and angle theta

 

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.

  1. soh = sine of theta is equal to opposite side over the hypotenuse.
  2. cah = cosine of theta is equal to adjacent side over the hypotenuse.
  3. toa = tangent of theta is equal to opposite side over the adjacent side.
  4. cho = cosecant of theta is equal to hypotenuse over the opposite side.
  5. sha = secant of theta is equal to hypotenuse over the adjacent side.
  6. cao = cotangent of theta is equal to adjacent side over the opposite 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 right triangle

 

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,
 

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

 

Derivation of Pythagorean Theorem

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

 

The Oblique Triangle

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

oblique-triangle-abc.gif

 

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...
 

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

 

 

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.
 

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

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

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

 

 

Video