Trig Identities Lesson - Unit 3 MK

Trig Identities

+ Reciprocal
+ Quotient

Trig Identities

+ Pythagorean
+ Addition/Subtraction

Reciprocal Identities

• Understand and be ready to apply the reciprocal properties including sine, cosine, tangent, cosecant, secant, and cotegent with numerator is 1

• Be aware of the relationship between sine and cosecant, between cosine and secant, and between tangent and cotangent

Pythagorean Identities

• Derive the Pythagorean identities from the unit circle definition of sine and cosine.

• Be aware of the relationship between sine and cosecant, between cosine and secant, and between tangent and cotangent

• Verify trigonometric identities using Pythagorean relationships.

• Solve real-world problems involving right triangles and periodic phenomena by employing Pythagorean identities.

Multiple Choice

reciprocal of cos(θ) is?

• sin(θ)

• csc(θ)

• sec(θ)

• cot(θ)

• tan(θ)

Multiple Select

$$\sin^2\theta=$$  

• $$\frac{1}{\csc^2\theta}$$  

• $$1-\cos^2\theta$$

• $$1-\sin^2\theta$$

• $$\frac{1}{\sec^2\theta}$$

Multiple Choice

reciprocal of csc(θ) is?

• sin(θ)

• cos(θ)

• sec(θ)

• cot(θ)

• tan(θ)

Multiple Select

$$\cos^2\theta=$$  

• $$\frac{1}{\csc^2\theta}$$  

• $$1-\cos^2\theta$$

• $$1-\sin^2\theta$$

• $$\frac{1}{\sec^2\theta}$$

Multiple Choice

reciprocal of cot(θ) is?

• sin(θ)

• csc(θ)

• sec(θ)

• sin(θ)

• tan(θ)

Multiple Select

$$\cot^2\theta=$$  

• $$\frac{1}{\tan^2\theta}$$  

• $$\csc^2\theta\ -1$$

• $$\sin^2\theta-1$$

• $$\frac{\sin^2\theta}{\cos^2\theta}$$

• $$\frac{\cos^2\theta}{\sin^2\theta}$$

Multiple Choice

reciprocal of sec(θ) is?

• cos(θ)

• csc(θ)

• sin(θ)

• tan(θ)

• cot(θ)

Multiple Select

$$\tan^2\theta=$$  

• $$\frac{1}{\cot^2\theta}$$  

• $$\sec^2\theta\ -1$$

• $$\sin^2\theta-1$$

• $$\frac{\sin^2\theta}{\cos^2\theta}$$

• $$\frac{\cos^2\theta}{\sin^2\theta}$$

Multiple Choice

reciprocal of sin(θ) is?

• cos(θ)

• csc(θ)

• sec(θ)

• tan(θ)

• cot(θ)

Multiple Select

$$\sec^2\theta=$$  

• $$\frac{1}{\sin^2\theta}$$  

• $$\frac{1}{\cos^2\theta}$$

• $$1+\tan^2\theta$$

• $$1+\cot^2\theta$$

Multiple Choice

reciprocal of tan(θ) is?

• cos(θ)

• csc(θ)

• sec(θ)

• sin(θ)

• cot(θ)

Multiple Select

$$\csc^2\theta=$$  

• $$\frac{1}{\sin^2\theta}$$  

• $$\frac{1}{\cos^2\theta}$$

• $$1+\tan^2\theta$$

• $$1+\cot^2\theta$$

Multiple Choice

Given cos(60⁰) = 1/2, evaluate  sec(60⁰)=?

• $$\frac{1}{2}$$  

• $$\frac{\sqrt{3}}{2}$$  

• $$\frac{\sqrt{2}}{2}$$  

• 1

• 2

Multiple Select

$$1=$$  

• $$\sec^2\theta-\tan^2\theta$$  

• $$\csc^2\theta-\cot^2\theta$$

• $$\sin^2\theta\ +\cos^2\theta$$

• $$\sin^2\theta-\cot^2\theta$$

• $$\cos^2\theta-\tan^2\theta$$

Multiple Choice

Given sin(30⁰) = 1/2, evaluate  csc(30⁰)=?

• $$\frac{1}{2}$$  

• $$\frac{\sqrt{3}}{2}$$  

• $$\frac{\sqrt{2}}{2}$$  

• 1

• 2

Objectives

• We will be able to expand a trig expression by applying the sum or difference identity of that trig function.

• We will be able to condense the expanded form of the sum or difference identity.

• We will be able to evaluate the sin, cos, or tan of unfamiliar angles using the sum of difference of familiar angles.

Multiple Choice

Given $$\csc\left(60\degree\right)=\frac{2\sqrt[]{3}}{3}$$ , evaluate   $$\sin\left(60\degree\right)=?$$

• $$\frac{1}{2}$$  

• $$\frac{\sqrt{3}}{2}$$  

• $$\frac{\sqrt{2}}{2}$$  

• 1

• 2

A rule to split angle is that angle must be equal to the sum or difference of two angles of reference (30⁰, 60⁰, 90⁰, 180⁰, 270⁰, and 360⁰)

Multiple Choice

Given $$\sec\left(30\degree\right)=\frac{2\sqrt[]{3}}{3}$$ , evaluate   $$\cos\left(30\degree\right)=?$$

• $$\frac{1}{2}$$  

• $$\frac{\sqrt{3}}{2}$$  

• $$\frac{\sqrt{2}}{2}$$  

• 1

• 2

Multiple Choice

According to the reference of angle rule, which of the followings is best equivalent to sin(105⁰)?

• sin(45⁰+60⁰)

• sin(180⁰-75⁰)

• sin(150⁰+45⁰)

• sin(15⁰+90⁰)

Multiple Choice

Given $$\tan\left(30\degree\right)=\frac{\sqrt[]{3}}{3}$$ , evaluate   $$\cot\left(30\degree\right)=?$$

• 0

• $$\frac{\sqrt{3}}{3}$$  

• $$\sqrt{3}$$  

• 1

• undefined

Multiple Select

According to the reference of angle rule, which of the followings is best equivalent to cos(150⁰)?

• cos(75⁰+75⁰)

• cos(180⁰-30⁰)

• cos(100⁰+50⁰)

• cos(60⁰+90⁰)

Multiple Choice

Given $$\cot\left(60\degree\right)=\frac{\sqrt[]{3}}{3}$$ , evaluate   $$\tan\left(60\degree\right)=?$$

• 0

• $$\frac{\sqrt{3}}{3}$$  

• $$\sqrt{3}$$  

• 1

• undefined

Multiple Select

According to the reference of angle rule, which of the followings is best equivalent to tan(300⁰)?

• cos(270⁰+30⁰)

• cos(180⁰-120⁰)

• cos(90⁰+210⁰)

• cos(360⁰-60⁰)

Quotient Identities

• Understand and be ready to apply the quotient properties including tangent and cotegent relationship with sine and cosine

• Be aware of tangent and cotangent have both reciprocal and quotient properties

Sum and Difference Identities

Categorize

Options (4)

$$\sin A\cos B+\cos A\sin B$$  

$$\sin A\cos B-\cos A\sin B$$  

$$\cos A\cos B\ -\sin A\sin B$$  

$$\cos A\cos B+\sin A\sin B$$  

Match the sum and difference identities for sine and cosine

Sin(A + B)

Sin(A - B)

Cos(A + B)

Cos (A - B)

Multiple Select

Which are equivalent to tan(θ)?

• 1/cot(θ)

• sin(θ)/cos(θ)

• cos(θ)/sin(θ)

• 1/tan(θ)

• csc(θ)/sec(θ)

Multiple Select

$$\tan\left(A+B\right)=$$

• $$\frac{\tan\left(A\right)+\tan\left(B\right)}{\left(1-\tan A\tan B\right)}$$

• $$\frac{\tan\left(A\right)-\tan\left(B\right)}{\left(1+\tan A\tan B\right)}$$

• $$\frac{\sin\left(A+B\right)}{\cos\left(A+B\right)}$$

• $$\frac{\sin\left(A-B\right)}{\cos\left(A-B\right)}$$

Multiple Select

Which are equivalent to cot(θ)?

• 1/cot(θ)

• sin(θ)/cos(θ)

• cos(θ)/sin(θ)

• 1/tan(θ)

• csc(θ)/sec(θ)

Multiple Select

$$\tan\left(A-B\right)=$$

• $$\frac{\tan\left(A\right)+\tan\left(B\right)}{\left(1-\tan A\tan B\right)}$$

• $$\frac{\tan\left(A\right)-\tan\left(B\right)}{\left(1+\tan A\tan B\right)}$$

• $$\frac{\sin\left(A-B\right)}{\cos\left(A-B\right)}$$

• $$\frac{\sin\left(A-B\right)}{\cos\left(A-B\right)}$$

Multiple Choice

According to the reciprocal identity, which proof of the followings is sinx = 1/cscx

• $$\sin x=1\times\frac{(\sin x}{1})$$

  $$=\sin x$$

• $$1/\csc x=1/(1/\sin x)$$

  $$=1/\csc x$$

• $$\sin x\ =1\div\frac{1}{\sin x}$$

  $$=\ 1\div\csc x=\ \frac{1}{\csc x}$$

• $$\sin x=\sin x\div\left(\frac{\sin x}{\sin x}\right)$$ $$=\ \sin x\div\left(\sin x\times\csc x\right)$$ $$=\csc x$$

Multiple Choice

$$\cos90^o\cos45^o-\sin90^{o\ }\sin45^o$$  is equivalent to

• $$\sin\ 45^{o\ }$$  

• $$\sin\ 135\degree$$  

• $$\cos\ 135^{o\ }$$  

• $$\cos45^{o\ }$$  

Multiple Choice

According to the reciprocal identity, which proof of the followings is $$\cos x\ =\frac{1}{\sec x}$$

• $$\cos x=1\times\frac{(\cos x}{1})$$

  $$=1\div\frac{1}{\cos x}$$

  $$=1\div\sec x\ =\ \frac{1}{\sec x}$$

• $$1/\sec x=1/(1/\cos x)$$

  $$=1/\sec x$$

• $$\cos x\ =1\times\frac{1}{\cos x}$$

  $$=\ 1\div\sec x=\ \frac{1}{\sec x}$$

• $$\cos x=\cos x\div\left(\frac{\cos x}{\cos x}\right)$$ $$=\ \cos x\div\left(\cos x\times\sec x\right)$$ $$=\sec x$$

Multiple Choice

$$\cos90^o\cos45^o+\sin90^{o\ }\sin45^o$$  is equivalent to

• $$\sin\ 45^{o\ }$$  

• $$\sin\ 135\degree$$  

• $$\cos\ 135^{o\ }$$  

• $$\cos45^{o\ }$$  

Multiple Choice

According to the reciprocal identity, which proof of the followings is $$\cot x\ =\frac{1}{\tan x}$$

• $$\cot x=1\times\frac{(\cot x}{1})$$

  $$=1\times\frac{1}{\cot x}$$

  $$=\tan$$

• $$1/\tan x=1\div\tan x$$

  $$=1\div\left(\frac{\sin x}{\cos x}\right)$$

  $$1\times\frac{\cos x}{\sin x}=\cot x$$

• $$\cot x\ =1\times\frac{1}{\cot x}$$

  $$=\ 1\div\cot x=\ \frac{1}{\cot x}$$

• $$\cot x=\cot x\div\left(\frac{\cot x}{\cot x}\right)$$ $$=\ \cot x\div\left(\frac{1}{\cot x\left(\tan x\right)}\right)$$ $$=\tan x$$

Multiple Choice

$$\sin90^o\cos45^o+\cos90^{o\ }\sin45^o$$  is equivalent to

• $$\sin\ 45^{o\ }$$  

• $$\sin\ 135\degree$$  

• $$\cos\ 135^{o\ }$$  

• $$\cos45^{o\ }$$  

Multiple Choice

According to quotient identity we have tanx = sinx/cosx, tanx = 1/cotx, show that cotx = cosx/sinx.

• True because

  $$\cot x=\frac{1}{\tan x}$$

  $$1\div\left(\frac{\sin x}{\cos x}\right)\ =\ \frac{\cos x}{\sin x}$$

• False because

  $$\tan x\ =\ \frac{\sin x}{\cos x},\ then$$

  $$\frac{1}{\tan x}=\ \frac{\cos x}{\sin x}$$

• False because

  $$\cot x\ =\ 1\div\tan x=\ \frac{\sin x}{\cos x}$$

• True because

  $$\frac{\cos x}{\sin x}=1\times\frac{\sin x}{\cos x}$$ $$=\frac{1}{\tan x}=\cot x$$

Multiple Choice

$$\sin90^o\cos45^o-\cos90^{o\ }\sin45^o$$  is equivalent to

• $$\sin\ 45^{o\ }$$  

• $$\sin\ 135\degree$$  

• $$\cos\ 135^{o\ }$$  

• $$\cos45^{o\ }$$  

Fill in the Blanks

Given sec(θ) = 7, evaluate cos(θ) = ?

Multiple Choice

Which of the followings is more acurate proof to determine cos(180⁰ + θ)?

• sin(180⁰)cosθ + cos(180⁰)sinθ =

  -sinθ

• sin(180⁰)cosθ -cos(180⁰)sinθ =

  sinθ

• cos(180⁰)cosθ -sin(180⁰)sinθ =

  -cosθ

• cos(180⁰)cosθ + sin(180⁰)sinθ =

  -cosθ

Fill in the Blanks

Given csc(θ) = 5, evaluate sin(θ) = ?

Multiple Choice

Which of the followings is more acurate proof to determine sin(180⁰ + θ)?

• sin(180⁰)cosθ + cos(180⁰)sinθ =

  -sinθ

• sin(180⁰)cosθ -cos(180⁰)sinθ =

  sinθ

• cos(180⁰)cosθ -sin(180⁰)sinθ =

  -cosθ

• cos(180⁰)cosθ + sin(180⁰)sinθ =

  -cosθ

Multiple Choice

Give sin(θ) = 0, cos(θ) = -1 Calculate exactvalue of sin(θ + φ)

• sin(θ)cos(φ) + cos(θ)sin(φ)

  = sin(φ)

• sin(θ)cos(φ) + cos(θ)sin(φ)

  = -sin(φ)

• sin(θ)cos(φ) - cos(θ)sin(φ)

  = sin(θ)

• sin(θ)cos(φ) - cos(θ)sin(φ)

  = -sin(θ)