Solving Trigonometry: 2011

4.3 Equivalent Trigonometric Expressions

These trigonometric identities will help you understand how equations are equal and you will apply your understanding of these trigonometric identities in questions asking for you to use equivalent trigonometric expressions.
The identities mostly used in this section are Co function identities for angles in the 1st and 2nd Quadrant.
This means that you will use a specific trigonometric identity for the angle found in either the 1st or 2nd quadrant.

1st. Quadrant:
1. sin30º= cos60º, both approx. = 0.5000
2. sin50º= cos40º, both approx. = 0.7660

2nd. Quadrant:
1. -sin20º=cos110º, both approx. = -0.3420
2. sin100º=cos10º, both approx. = -0.1736
Key: degrees is substituted for the x value

We will come up with the following identities:
1st Quadrant:
1. sinx = cos((π/2)-x)
2. cosx = sin((π/2)-x)

2nd Quadrant:
1. sinx= cos(x+(π/2))
2. -cosx= sin(x+(π/2))

3rd Quadrant:

1. -sinx=cos(x+π)

2. -cosx=sin(x+π)

4rd Quadrant:

1. -sinx=cos(2π-x)

2. cosx=sin(2π-x)

Pythagorean Identities

For angle θ at which the functions are defined:

(1) cos²θ + sin²θ = 1
(2) 1 + tan²θ = sec²θ

(3) 1 + cot²θ = csc²θ

Dividing both sides of the first equation by cos²θ results in the second equation. Similarly, dividing both sides of the first equation by sin²θ results in the third equation.

Reciprocal Identities

Cofunction Identities

Double-Angle Identities

(1) sin

2

θ = 2 sin θ cos θ

(2) cos

2

θ = 2 cos²θ

-

1
(3) cos

2

θ = 1

-

2 sin²θ

(4) cos²θ = ½(1 + cos

2

θ)
(5) sin²θ = ½(1

-

cos

2

θ)

Sum & Difference Identities

sin(α + β) = sin α cos β + sin β cos α

sin(α

-

β) = sin α cos β

-

sin β cos α

cos(α + β) = cos α cos β

-

sin α sin β

cos(α

-

β) = cos α cos β

+

sin α sin β

Even-Odd Identities

4.2 Trigonometric Ratios and Special Angles

Trigonometric Ratios

The relationships between the angles and the sides of a right triangle are expressed in terms of TRIGONOMETRIC RATIOS.

Figure 1. A right angle triangle

A right triangle has one of its interior angles measuring 90° (a right angle).
Right triangles obey the Pythagorean theorem: the sum of the squares of the lengths of the two legs is equal to the square of the length of the hypotenuse: a² + b² = c², where a and b are the lengths of the legs and c is the length of the hypotenuse.

In a right triangle, the six trigonometric ratios are;

sine
cosine
tangent
cosecant
secant
cotangent

Figure 2. Hypotenuse, Opposite and Adjacent

Sine function (sin), defined as the ratio of the side opposite the angle to the hypotenuse.

$\sin A=\frac{\textrm{opposite}}{\textrm{hypotenuse}}=\frac{a}{\,c\,}\,.$

Cosine function (cos), defined as the ratio of the adjacent leg to the hypotenuse.

$\cos A=\frac{\textrm{adjacent}}{\textrm{hypotenuse}}=\frac{b}{\,c\,}\,.$

Tangent function (tan), defined as the ratio of the opposite leg to the adjacent leg.

$\tan A=\frac{\textrm{opposite}}{\textrm{adjacent}}=\frac{a}{\,b\,}=\frac{\sin A}{\cos A}\,.$

Cosecant csc(A), or cosec(A), is the reciprocal of sin(A), i.e. the ratio of the length of the hypotenuse to the length of the opposite side:

$\csc A = \frac {1}{\sin A} = \frac {\textrm{hypotenuse}} {\textrm{opposite}} = \frac {h} {a}.$

Secant sec(A) is the reciprocal of cos(A), i.e. the ratio of the length of the hypotenuse to the length of the adjacent side:

$\sec A = \frac {1}{\cos A} = \frac {\textrm{hypotenuse}} {\textrm{adjacent}} = \frac {h} {b}.$

Cotangent cot(A) is the reciprocal of tan(A), i.e. the ratio of the length of the adjacent side to the length of the opposite side:

$\cot A = \frac {1}{\tan A} = \frac {\textrm{adjacent}} {\textrm{opposite}} = \frac {b} {a}.$

Special Angles

The values of trig functions of specific angles can be represented by known ratios, and are good to commit to memory to help you work through problems faster.

These 'special' angles can be remembered by examining two different triangles.

Specifically, the trig functions for angles of 0, 30, 45, 60, 90 degrees are the special ones.