Trigonometry 1 Overview

Trigonometry is the branch of mathematics dealing with the relationship of sides and angles of triangles.

Subjects > Mathematics > Trigonometry 1

Before moving direct into the problems of Trigonometry 1, let's see what we need to know about Trigonometry.

Right Triangle Definitions of Trigonometric Functions

Consider the following diagram.

wesovu Trigonomtry Triangle 1
Adjacent is the side adjacent to the angle in consideration. So if we are considering Angle `A`, then the adjacent side is `CB`

let use `opp` = opposite, `hyp` = hypotenuse and `adj` = adjacent
From the diagram above,

`sin theta = (opp)/(hyp) = y/r`

`cos theta = (adj)/(hyp) = x/r`

`tan theta = (opp)/(adj) = y/x`

`csc theta = (hyp)/(opp) = r/y`

`sec theta = (hyp)/(adj) = r/x`

`cot theta = (adj)/(opp) = x/y`

`csc theta = 1/(sin theta)`

`sec theta = 1/(cos theta)`

`cot theta = 1/(tan theta)`

Trigonometric Values of some Special Angles

Degrees`0^0``30^0``45^0``60^0``90^0``180^0``270^0`
Radians 0`(pi)/6``(pi)/4``(pi)/3``(pi)/2``pi``(3pi)/2`
`sin theta` `0``1/2``(sqrt(2))/2``(sqrt(3))/2``1``0``-1`
`cos theta` `1``(sqrt(3))/2``(sqrt(2))/2``1/2``0``-1``0`
`tan theta` `0``(sqrt(3))/3``1``sqrt(3)`undefined`0`undefined

Angles of Elevation and Depression

Angle of Elevation
wesovu Angle of Elevation
Angle of elevation of an object as seen by an observer is the angle between the horizontal and the line from the object to the observer's eye (line of sight)

Angle of Depression
wesovu Angle of depression
If the object is below the level of the observer, then the angle between the horizontal and the observer's line of sight is called the angle of depression

The Unit Circle

Unit circle is a circle with radius of 1, and it is used to show certain common angles.

wesovu Unit Circle
The unit circle, showing coordinates and angle measures of certain points.

Trigonometry 1 Identities

Pythagorean Identities
`sin^2 x + cos^2 x = 1`
`1 + tan^2 x = sec^2 x`
`1 + cot^2 x = csc^2 x`

Pythagorean Identities in Radical Form
`sin x = pm sqrt(1 - cos^2 x)`
`tan x = pm sqrt(sec^2 x - 1)`
`cos x = pm sqrt(1 - sin^2 x)`
`cot x = pm sqrt(csc^2 x - 1)`

Odd Identities
`sin (-x) = -sin x`
`csc (-x) = -csc x`
`tan (-x) = -tan x`
`cot (-x) = -cot x`

Even Identities
`cos (-x) = cos x`
`sec (-x) = sec x`

Sum and Difference Formulas/Identities
`sin (A + B) = sin Acos B + cos Asin B`
`sin (A - B) = sin Acos B - cos Asin B`

`cos (A + B) = cos Acos B - sin Asin B`
`cos (A - B) = cos Acos B + sin Asin B`

`tan (A + B) = (tan A + tan B)/(1 - tan Atan B)`
`tan (A - B) = (tan A - tan B)/(1 - tan Atan B)`

Cofunction Identities
`sin ((pi)/2 - x) = cos x`
`cos ((pi)/2 - x) = sin x`
`tan ((pi)/2 - x) = cot x`
`cot ((pi)/2 - x) = tan x`
`sec ((pi)/2 - x) = csc x`
`csc ((pi)/2 - x) = sec x`

Half Angle Identities
`sin A/2 = pm sqrt((1 - cos A)/2)`

`cos A/2 = pm sqrt((1 + cos A)/2)`

`tan A/2 = (1 - cos A)/(sin A)`

`tan A/2 = (sin A)/(1 + cos A)`

You can learn more about Trigonometry 2 here.

Now, let's move in. Solved Trigonometry questions

Recommended Lessons

Before moving direct into the problems of Trigonometry 1, let's see what we need to know about Trigonometry.

Right Triangle Definitions of Trigonometric Functions

Consider the following diagram.

wesovu Trigonomtry Triangle 1
Adjacent is the side adjacent to the angle in consideration. So if we are considering Angle `A`, then the adjacent side is `CB`

let use `opp` = opposite, `hyp` = hypotenuse and `adj` = adjacent
From the diagram above,

`sin theta = (opp)/(hyp) = y/r`

`cos theta = (adj)/(hyp) = x/r`

`tan theta = (opp)/(adj) = y/x`

`csc theta = (hyp)/(opp) = r/y`

`sec theta = (hyp)/(adj) = r/x`

`cot theta = (adj)/(opp) = x/y`

`csc theta = 1/(sin theta)`

`sec theta = 1/(cos theta)`

`cot theta = 1/(tan theta)`

Trigonometric Values of some Special Angles

Degrees`0^0``30^0``45^0``60^0``90^0``180^0``270^0`
Radians 0`(pi)/6``(pi)/4``(pi)/3``(pi)/2``pi``(3pi)/2`
`sin theta` `0``1/2``(sqrt(2))/2``(sqrt(3))/2``1``0``-1`
`cos theta` `1``(sqrt(3))/2``(sqrt(2))/2``1/2``0``-1``0`
`tan theta` `0``(sqrt(3))/3``1``sqrt(3)`undefined`0`undefined

Angles of Elevation and Depression

Angle of Elevation
wesovu Angle of Elevation
Angle of elevation of an object as seen by an observer is the angle between the horizontal and the line from the object to the observer's eye (line of sight)

Angle of Depression
wesovu Angle of depression
If the object is below the level of the observer, then the angle between the horizontal and the observer's line of sight is called the angle of depression

The Unit Circle

Unit circle is a circle with radius of 1, and it is used to show certain common angles.

wesovu Unit Circle
The unit circle, showing coordinates and angle measures of certain points.

Trigonometry 1 Identities

Pythagorean Identities
`sin^2 x + cos^2 x = 1`
`1 + tan^2 x = sec^2 x`
`1 + cot^2 x = csc^2 x`

Pythagorean Identities in Radical Form
`sin x = pm sqrt(1 - cos^2 x)`
`tan x = pm sqrt(sec^2 x - 1)`
`cos x = pm sqrt(1 - sin^2 x)`
`cot x = pm sqrt(csc^2 x - 1)`

Odd Identities
`sin (-x) = -sin x`
`csc (-x) = -csc x`
`tan (-x) = -tan x`
`cot (-x) = -cot x`

Even Identities
`cos (-x) = cos x`
`sec (-x) = sec x`

Sum and Difference Formulas/Identities
`sin (A + B) = sin Acos B + cos Asin B`
`sin (A - B) = sin Acos B - cos Asin B`

`cos (A + B) = cos Acos B - sin Asin B`
`cos (A - B) = cos Acos B + sin Asin B`

`tan (A + B) = (tan A + tan B)/(1 - tan Atan B)`
`tan (A - B) = (tan A - tan B)/(1 - tan Atan B)`

Cofunction Identities
`sin ((pi)/2 - x) = cos x`
`cos ((pi)/2 - x) = sin x`
`tan ((pi)/2 - x) = cot x`
`cot ((pi)/2 - x) = tan x`
`sec ((pi)/2 - x) = csc x`
`csc ((pi)/2 - x) = sec x`

Half Angle Identities
`sin A/2 = pm sqrt((1 - cos A)/2)`

`cos A/2 = pm sqrt((1 + cos A)/2)`

`tan A/2 = (1 - cos A)/(sin A)`

`tan A/2 = (sin A)/(1 + cos A)`

You can learn more about Trigonometry 2 here.

Now, let's move in. Solved Trigonometry questions

Recommended Lessons