Skip to content Skip to navigation

OpenStax-CNX

You are here: Home » Content » Trigonometry: Trig identities (Grade 11)

Navigation

Lenses

What is a lens?

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

This content is ...

Affiliated with (What does "Affiliated with" mean?)

This content is either by members of the organizations listed or about topics related to the organizations listed. Click each link to see a list of all content affiliated with the organization.
  • FETMaths display tagshide tags

    This module is included inLens: Siyavula: Mathematics (Gr. 10-12)
    By: Siyavula

    Review Status: In Review

    Click the "FETMaths" link to see all content affiliated with them.

    Click the tag icon tag icon to display tags associated with this content.

  • Bookshare

    This module is included inLens: Bookshare's Lens
    By: Bookshare - A Benetech InitiativeAs a part of collection: "FHSST: Grade 11 Maths"

    Comments:

    "Accessible versions of this collection are available at Bookshare. DAISY and BRF provided. "

    Click the "Bookshare" link to see all content affiliated with them.

  • Siyavula: Mathematics display tagshide tags

    This module is included inLens: Siyavula Textbooks: Maths
    By: Free High School Science Texts ProjectAs a part of collection: "FHSST: Grade 11 Maths"

    Click the "Siyavula: Mathematics" link to see all content affiliated with them.

    Click the tag icon tag icon to display tags associated with this content.

Recently Viewed

This feature requires Javascript to be enabled.

Tags

(What is a tag?)

These tags come from the endorsement, affiliation, and other lenses that include this content.
 

Trigonometry: Trig identities (Grade 11)

Module by: Free High School Science Texts Project. E-mail the author

Trigonometric Identities

Deriving Values of Trigonometric Functions for 3030, 4545 and 6060

Keeping in mind that trigonometric functions apply only to right-angled triangles, we can derive values of trigonometric functions for 3030, 4545 and 6060. We shall start with 4545 as this is the easiest.

Take any right-angled triangle with one angle 4545. Then, because one angle is 9090, the third angle is also 4545. So we have an isosceles right-angled triangle as shown in Figure 1.

Figure 1: An isosceles right angled triangle.
Figure 1 (MG11C17_019.png)

If the two equal sides are of length aa, then the hypotenuse, hh, can be calculated as:

h 2 = a 2 + a 2 = 2 a 2 h = 2 a h 2 = a 2 + a 2 = 2 a 2 h = 2 a
(1)

So, we have:

sin ( 45 ) = opposite ( 45 ) hypotenuse = a 2 a = 1 2 sin ( 45 ) = opposite ( 45 ) hypotenuse = a 2 a = 1 2
(2)
cos ( 45 ) = adjacent ( 45 ) hypotenuse = a 2 a = 1 2 cos ( 45 ) = adjacent ( 45 ) hypotenuse = a 2 a = 1 2
(3)
tan ( 45 ) = opposite ( 45 ) adjacent ( 45 ) = a a = 1 tan ( 45 ) = opposite ( 45 ) adjacent ( 45 ) = a a = 1
(4)

We can try something similar for 3030 and 6060. We start with an equilateral triangle and we bisect one angle as shown in Figure 2. This gives us the right-angled triangle that we need, with one angle of 3030 and one angle of 6060.

Figure 2: An equilateral triangle with one angle bisected.
Figure 2 (MG11C17_020.png)

If the equal sides are of length aa, then the base is 12a12a and the length of the vertical side, vv, can be calculated as:

v 2 = a 2 - ( 1 2 a ) 2 = a 2 - 1 4 a 2 = 3 4 a 2 v = 3 2 a v 2 = a 2 - ( 1 2 a ) 2 = a 2 - 1 4 a 2 = 3 4 a 2 v = 3 2 a
(5)

So, we have:

sin ( 30 ) = opposite ( 30 ) hypotenuse = a 2 a = 1 2 sin ( 30 ) = opposite ( 30 ) hypotenuse = a 2 a = 1 2
(6)
cos ( 30 ) = adjacent ( 30 ) hypotenuse = 3 2 a a = 3 2 cos ( 30 ) = adjacent ( 30 ) hypotenuse = 3 2 a a = 3 2
(7)
tan ( 30 ) = opposite ( 30 ) adjacent ( 30 ) = a 2 3 2 a = 1 3 tan ( 30 ) = opposite ( 30 ) adjacent ( 30 ) = a 2 3 2 a = 1 3
(8)
sin ( 60 ) = opposite ( 60 ) hypotenuse = 3 2 a a = 3 2 sin ( 60 ) = opposite ( 60 ) hypotenuse = 3 2 a a = 3 2
(9)
cos ( 60 ) = adjacent ( 60 ) hypotenuse = a 2 a = 1 2 cos ( 60 ) = adjacent ( 60 ) hypotenuse = a 2 a = 1 2
(10)
tan ( 60 ) = opposite ( 60 ) adjacent ( 60 ) = 3 2 a a 2 = 3 tan ( 60 ) = opposite ( 60 ) adjacent ( 60 ) = 3 2 a a 2 = 3
(11)

You do not have to memorise these identities if you know how to work them out.

Tip: Two useful triangles to remember:

Figure 3
Figure 3 (MG11C17_021.png)

Alternate Definition for tanθtanθ

We know that tanθtanθ is defined as: tanθ= opposite adjacent tanθ= opposite adjacent This can be written as:

tan θ = opposite adjacent × hypotenuse hypotenuse = opposite hypotenuse × hypotenuse adjacent tan θ = opposite adjacent × hypotenuse hypotenuse = opposite hypotenuse × hypotenuse adjacent
(12)

But, we also know that sinθsinθ is defined as: sinθ= opposite hypotenuse sinθ= opposite hypotenuse and that cosθcosθ is defined as: cosθ= adjacent hypotenuse cosθ= adjacent hypotenuse

Therefore, we can write

tan θ = opposite hypotenuse × hypotenuse adjacent = sin θ × 1 cos θ = sin θ cos θ tan θ = opposite hypotenuse × hypotenuse adjacent = sin θ × 1 cos θ = sin θ cos θ
(13)

Tip:

tanθtanθ can also be defined as: tanθ=sinθcosθtanθ=sinθcosθ

A Trigonometric Identity

One of the most useful results of the trigonometric functions is that they are related to each other. We have seen that tanθtanθ can be written in terms of sinθsinθ and cosθcosθ. Similarly, we shall show that: sin2θ+cos2θ=1sin2θ+cos2θ=1

We shall start by considering ABCABC,

Figure 4
Figure 4 (MG11C17_022.png)

We see that: sinθ=ACBCsinθ=ACBC and cosθ=ABBC.cosθ=ABBC.

We also know from the Theorem of Pythagoras that: AB2+AC2=BC2.AB2+AC2=BC2.

So we can write:

sin 2 θ + cos 2 θ = A C B C 2 + A B B C 2 = A C 2 B C 2 + A B 2 B C 2 = A C 2 + A B 2 B C 2 = B C 2 B C 2 ( from Pythagoras ) = 1 sin 2 θ + cos 2 θ = A C B C 2 + A B B C 2 = A C 2 B C 2 + A B 2 B C 2 = A C 2 + A B 2 B C 2 = B C 2 B C 2 ( from Pythagoras ) = 1
(14)

Exercise 1: Trigonometric Identities A

Simplify using identities:

  1. tan 2 θ · cos 2 θ tan 2 θ · cos 2 θ
  2. 1 cos 2 θ - tan 2 θ 1 cos 2 θ - tan 2 θ
Solution
  1. Step 1. Use known identities to replace tanθtanθ :
    = tan 2 θ · cos 2 θ = sin 2 θ cos 2 θ · cos 2 θ = sin 2 θ = tan 2 θ · cos 2 θ = sin 2 θ cos 2 θ · cos 2 θ = sin 2 θ
    (15)
  2. Step 2. Use known identities to replace tanθtanθ :
    = 1 cos 2 θ - tan 2 θ = 1 cos 2 θ - sin 2 θ cos 2 θ = 1 - sin 2 θ cos 2 θ = cos 2 θ cos 2 θ = 1 = 1 cos 2 θ - tan 2 θ = 1 cos 2 θ - sin 2 θ cos 2 θ = 1 - sin 2 θ cos 2 θ = cos 2 θ cos 2 θ = 1
    (16)

Exercise 2: Trigonometric Identities B

Prove: 1-sinxcosx=cosx1+sinx1-sinxcosx=cosx1+sinx

Solution
  1. Step 1. Use trig identities :
    LHS = 1 - sin x cos x = 1 - sin x cos x × 1 + sin x 1 + sin x = 1 - sin 2 x cos x ( 1 + sin x ) = cos 2 x cos x ( 1 + sin x ) = cos x 1 + sin x = RHS LHS = 1 - sin x cos x = 1 - sin x cos x × 1 + sin x 1 + sin x = 1 - sin 2 x cos x ( 1 + sin x ) = cos 2 x cos x ( 1 + sin x ) = cos x 1 + sin x = RHS
    (17)

Trigonometric identities

  1. Simplify the following using the fundamental trigonometric identities:
    1. cosθtanθcosθtanθ
    2. cos2θ.tan2θ+tan2θ.sin2θcos2θ.tan2θ+tan2θ.sin2θ
    3. 1-tan2θ.sin2θ1-tan2θ.sin2θ
    4. 1-sinθ.cosθ.tanθ1-sinθ.cosθ.tanθ
    5. 1-sin2θ1-sin2θ
    6. 1-cos2θcos2θ-cos2θ1-cos2θcos2θ-cos2θ
  2. Prove the following:
    1. 1+sinθcosθ=cosθ1-sinθ1+sinθcosθ=cosθ1-sinθ
    2. sin2θ+(cosθ-tanθ)(cosθ+tanθ)=1-tan2θsin2θ+(cosθ-tanθ)(cosθ+tanθ)=1-tan2θ
    3. (2cos2θ-1)1+1(1+tan2θ)=1-tan2θ1+tan2θ(2cos2θ-1)1+1(1+tan2θ)=1-tan2θ1+tan2θ
    4. 1cosθ-cosθtan2θ1=11cosθ-cosθtan2θ1=1
    5. 2sinθcosθsinθ+cosθ=sinθ+cosθ-1sinθ+cosθ2sinθcosθsinθ+cosθ=sinθ+cosθ-1sinθ+cosθ
    6. cosθsinθ+tanθ·cosθ=1sinθcosθsinθ+tanθ·cosθ=1sinθ

Content actions

Download module as:

Add module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks