Pre$Calculus+Mathematics+12+–+6.1...

Pre-‐Calculus Mathematics 12 – 6.1 – Trigonometric Identities and Equations

The FUNdamental TRIGonometric Identities

In trigonometry, there are expressions and equations that are true for any given angle. These are called identities. An infinite number of trigonometric identities exist, and we are going to prove many of these identities, but we are going to need some basic identities first. The six basic trig ratios will lead to our first identities

sin θ = cos θ = tan θ =

csc θ = sec θ = cot θ =

And our knowledge of Pythagoras will determine the remaining FUNdamental TRIGonometric Identities

Goal: 1. Identify the Fundamental Trigonometric Identities 2. Simplify a Trigonometric Expression 3. Determine the restrictions on a Trigonometric Expression


Pythagorean Identities:

2 2sin cos 1θ θ+ = 2 21 tan secθ θ+ = 2 21 cot cscθ θ+ =

Reciprocal and Quotient Identities:

1seccos

θθ

= 1cscsin

θθ

= 1cottan

θθ

= sintancos

θθ

θ=

coscotsin

θθ

θ=

Corollary Identities ( a statement that follows readily from a previous statement)

2 2sin cos 1θ θ+ = 2 21 tan secθ θ+ = 2 21 cot cscθ θ+ =

Simplifying a Trigonometric Expression There are many different strategies to simplifying a trigonometric expression. The following examples will look at the most common types of strategies.

Write as a fraction with a common denominator

cos𝜃 +sin𝜃cos𝜃

Factor as a difference of squares

1− 𝑐𝑜𝑠! 𝜃 𝑠𝑒𝑐!𝜃 − 𝑡𝑎𝑛!𝜃

Change everything to sine and cosine.

tan𝜃 sec𝜃

Multiply by the conjugate

1

1− sin𝜃


Now we can use these strategies along with the eight fundamental identities to simplify expressions Example 1: Simplify

sin𝜃 +𝑐𝑜𝑠!𝜃sin𝜃

Example 2: Simplify

1

𝑠𝑖𝑛! 𝜃 − 1

Example 3: Simplify

!! !"! !

!"# !!!"# !

Example 4: Simplify 𝑐𝑜𝑠!𝑥 − 2𝑐𝑜𝑠!𝑥 + 1

Example 5: Simplify !"#!!!

!"#!!!"#! !"#!

Example 6: Simplify !"# !

!!!"# !+ !"# !

!!!"# !


Restrictions Just like any algebraic expression, a trigonometric expression cannot have zero in the denominator. We must consider the exact values that would result in a denominator of zero. Example 7: Determine the restrictions on tan 𝑥 − csc 𝑥 for 0 ≤ 𝑥 < 2𝜋

Practice: Page 264 #2 -‐ 9 (as many as needed)

Pre-‐Calculus Mathematics 12 – 6.2 – Verifying Trigonometric Identities

Trigonometric Identities

When verifying trigonometric identities, the key is using the rules for algebra as well as the fundamental trigonometric identities to rewrite and simplify expressions. An identity has been proven when the right side of the equal sign is the same as the left side of the equal side.

Example 1: Prove the identity: !!!"#! !

!"#! != 𝑡𝑎𝑛!𝜃

Example 2: Prove the identity: !!!"#!!"#!

= sin 𝜃 + cos 𝜃

Goal: Verify and prove Trigonometric Identities


Example 3: Prove the identity: !"#$ ! !"#$ !"#! ! !

= cot 𝜃


Example 4: Prove the identity: sec 𝜃 csc 𝜃 tan 𝜃 = 𝑠𝑒𝑐!𝜃

Practice: Page 271 #1-‐ 26

Pre-‐Calculus Mathematics 12 – 6.3 –Trigonometric Equations

Trigonometric Equations

A trigonometric equation is different from a trigonometric identity in that the equation is true for some values of the variable and not all values. When solving a trig equation, we can solve for a specified domain of values ( usually 0 ≤ 𝜃 < 2𝜋 ) or in general form ( for all potential values)

Example 1: Solve: 𝑠𝑒𝑐 𝜃 = − !! for

a) 0 ≤ 𝜃 < 2𝜋 b) general form

Example 2: Solve: tan𝜃 = −0.3124 for a) 0 ≤ 𝜃 < 2𝜋 b) general form

Example 3: Solve: 2 𝑐𝑜𝑠!𝑥 − cos 𝑥 − 1 = 0 for a) 0 ≤ 𝑥 < 2𝜋 b) general form

Goal: 1. Solve trigonometric equations for conditional statements and general form 2. Solve equation with angles other then θ or x.

Pre-‐Calculus Mathematics 12 – 6.3 –Trigonometric Equations

Solving trigonometric equations with angles other than θ

Solving a trig equation for angles with coefficients ( 2θ, 3θ, etc) follows very similar steps: 1. Isolate the trig function. eg. sin 2θ 2. Determine the general solution(s) for angle 2θ 3. Determine the general solution(s) for angle θ. (divide by the coefficient) 4. Determine the specific (‘conditional’) solution(s) for the given interval

Example 4: Solve: 𝑠𝑖𝑛 2𝑥 = −1 for a) general form b) 0 ≤ 𝑥 < 2𝜋

Example 5: Solve: 3 𝑡𝑎𝑛 2𝜃 + 1 = 0 for a) general form b) 0 ≤ 𝜃 < 2𝜋

Example 6: Solve: 4 𝑠𝑖𝑛!2𝑥 + 2 sin 2𝑥 − 2 = 0 for a) 0 ≤ 𝜃 < 2𝜋 b) general form

Practice: Page 281 #1 -‐ 5

Pre-‐Calculus Mathematics 12 – 6.4 –Sum and Difference Identities

Sum and Difference Identities

Identities are not limited to the fundamental identities and single angles. We can also use identities involving sums and differences. The derivations of these identities are shown on page 287 of your textbook. We will be looking not at proving these identities but using these identities. ( )cos cos cos sin sinα + β = α β − α β ( )sin sin cos cos sinα + β = α β + α β

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

( ) ( )tan tan tan tantan tan

1 tan tan 1 tan tanα α

α + β α − β+ β = − β =

− α β + α β

Goal: 1. Use sum and difference identities to solve complex trig problems 2. Simplify expressions and prove identities involving sums and differences

Example 1: Find the exact value:

𝑠𝑖𝑛 !!cos !

!+ cos !

!sin !

!

Example 2: Find the exact value: cos 345°

Example 3: Express as a single function, then evaluate. 𝑐𝑜𝑠! !

!− 𝑠𝑖𝑛! !

!

Example 4: Express as a single function, then evaluate. !"#!! !"# !!!! !"#! !"#!!

Pre-‐Calculus Mathematics 12 – 6.4 –Sum and Difference Identities

Example 5: Given angle A in quadrant I and angle B in quadrant II, such that 𝑠𝑖𝑛𝐴 = !! 𝑎𝑛𝑑 cos𝐵 = − !"

!",

find 𝑡𝑎𝑛 𝐴 − 𝐵 .

Example 6: Prove: !"# !!!"# !!!!

= 𝑡𝑎𝑛 𝑥

Practice: Page 292 #1 -‐ 6

Pre-‐Calculus Mathematics 12 – 6.5– Double-‐Angle Identities

Double Angle Identities

Using the sum and difference identities, we can determine other trigonometric identities

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

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

( ) ( )tan tan tan tantan tan1 tan tan 1 tan tan

α αα + β α − β

+ β = − β =− α β + α β

Double Angle Identities (same angle)

2 2

2

2

cos2 cos sin2cos 11 2sin

θ θ θ

θ

θ

= −

= −

= −

tan 2𝜃 = ! !"#!!!!"# !!

Goal: 1. Identify the double angle trigonometric identities 2. Simplify and prove double-‐angle trigonometric expressions and equations

sin 2 2sin cosθ θ θ=

Pre-‐Calculus Mathematics 12 – 6.5– Double-‐Angle Identities

Example 3: Prove: !!!"# !!!"# !!

= !!!"# !!"# !!!"# !

Example 1: Simplify

21−cos4𝑥

Example 2: Solve, 0 ≤ 𝑥 ≤ 2𝜋 csc 𝑥 = 2 sec 2𝑥!

Practice: Page 300 #1 – 6 (as many as needed)

Pre$Calculus+Mathematics+12+–+6.1...

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