Solved Exercises on Trig Identities Intro in Algebra 2

Master Pythagorean and reciprocal trigonometric identities: sin²θ + cos²θ = 1, sec θ = 1/cos θ, and applications through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Pythagorean identity proof
Exercise 1
Prove the Pythagorean identity sin²θ + cos²θ = 1 using the unit circle definition.
Definition:

Unit circle: A circle with center (0,0) and radius 1. Any point (x,y) on the unit circle satisfies x² + y² = 1.

Proof method:
  1. Consider a point P(x,y) on the unit circle corresponding to angle θ
  2. Apply the definition of sine and cosine in terms of coordinates
  3. Use the equation of the unit circle
  4. Substitute and simplify
Unit Circle
x² + y² = 1
Definitions
x = cos θ, y = sin θ
Identity
cos²θ + sin²θ = 1
Step 1: Consider a point on the unit circle

For any angle θ, the point P(cos θ, sin θ) lies on the unit circle

Step 2: Apply the unit circle equation

Since P(x,y) is on the unit circle, x² + y² = 1

Step 3: Substitute coordinate definitions

Substituting x = cos θ and y = sin θ:

(cos θ)² + (sin θ)² = 1

Step 4: Simplify

cos²θ + sin²θ = 1

sin²θ + cos²θ = 1
Final answer:

The Pythagorean identity sin²θ + cos²θ = 1 is proven using the unit circle definition.

Applied rules:

• Unit circle equation: x² + y² = r² (with r = 1)

• Trigonometric definitions: cos θ = x, sin θ = y

• Algebraic substitution: Replace variables with their definitions

2 Reciprocal identity application
Exercise 2
Given that cos θ = 3/5, find the exact values of sec θ and sin θ using trigonometric identities.
Definition:

Reciprocal identities: sec θ = 1/cos θ, csc θ = 1/sin θ, cot θ = 1/tan θ

Given
cos θ = 3/5
Find
sec θ, sin θ
Identities
sec = 1/cos, sin²+cos²=1
Step 1: Find sec θ using reciprocal identity

sec θ = 1/cos θ = 1/(3/5) = 5/3

Step 2: Find sin θ using Pythagorean identity

sin²θ + cos²θ = 1

sin²θ + (3/5)² = 1

sin²θ + 9/25 = 1

sin²θ = 1 - 9/25 = 25/25 - 9/25 = 16/25

Step 3: Solve for sin θ

sin θ = ±√(16/25) = ±4/5

Step 4: Determine the sign

Without additional information about the quadrant, we have two possibilities:

sin θ = 4/5 or sin θ = -4/5

sec θ = 5/3
sin θ = ±4/5
Final answer:

sec θ = 5/3, sin θ = ±4/5

Applied rules:

• Reciprocal identity: sec θ = 1/cos θ

• Pythagorean identity: sin²θ + cos²θ = 1

• Sign determination: Requires quadrant information

3 Pythagorean identity manipulation
Exercise 3
Starting with sin²θ + cos²θ = 1, derive the identity tan²θ + 1 = sec²θ.
Definition:

Derived identity: An identity obtained by algebraic manipulation of a fundamental identity.

Start
sin²θ + cos²θ = 1
Divide by cos²θ
tan²θ + 1 = sec²θ
Result
tan²θ + 1 = sec²θ
Step 1: Start with the fundamental identity

sin²θ + cos²θ = 1

Step 2: Divide both sides by cos²θ

(sin²θ + cos²θ)/cos²θ = 1/cos²θ

Step 3: Separate the left side

sin²θ/cos²θ + cos²θ/cos²θ = 1/cos²θ

Step 4: Simplify each term

(sin θ/cos θ)² + 1 = (1/cos θ)²

tan²θ + 1 = sec²θ

Step 5: Verify the result

We have successfully derived: tan²θ + 1 = sec²θ

tan²θ + 1 = sec²θ
Final answer:

The identity tan²θ + 1 = sec²θ has been derived from sin²θ + cos²θ = 1.

Applied rules:

• Algebraic manipulation: Divide both sides by the same expression

• Definition substitution: tan θ = sin θ/cos θ, sec θ = 1/cos θ

• Identity derivation: Build new identities from fundamental ones

Trig Identities Fundamentals
sin²θ + cos²θ = 1
Fundamental Identity
Pythagorean
sin²θ + cos²θ = 1
Base identity
Reciprocal
sec θ = 1/cos θ
Definition
Derived
tan²θ + 1 = sec²θ
Manipulation
Key definitions:

Trigonometric identity: An equation involving trigonometric functions that is true for all values of the variable for which both sides are defined.

Pythagorean identities: Identities derived from the Pythagorean theorem.

Reciprocal identities: Identities expressing functions as reciprocals of others.

Pythagorean Identities: sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, 1 + cot²θ = csc²θ
Reciprocal Identities: sec θ = 1/cos θ, csc θ = 1/sin θ, cot θ = 1/tan θ
Tip 1: Always start with the more complex side when proving identities.
Tip 2: Convert everything to sine and cosine when stuck.
Tip 3: Remember: sin²θ means (sin θ)², not sin(θ²).
Tip 4: Check your work by substituting a specific angle.
Solution: Exercises 4 to 5
4 Complex identity verification
Exercise 4
Verify the identity: (sec θ - tan θ)(sec θ + tan θ) = 1
Definition:

Difference of squares: (a - b)(a + b) = a² - b² pattern often appears in trig identities.

Left Side
(sec θ - tan θ)(sec θ + tan θ)
Expand
sec²θ - tan²θ
Simplify
1
Step 1: Recognize the pattern

(sec θ - tan θ)(sec θ + tan θ) follows the difference of squares pattern: (a - b)(a + b) = a² - b²

Step 2: Apply the pattern

(sec θ - tan θ)(sec θ + tan θ) = sec²θ - tan²θ

Step 3: Apply the derived Pythagorean identity

From tan²θ + 1 = sec²θ, we get: sec²θ - tan²θ = 1

Step 4: Verify the result

Left side = sec²θ - tan²θ = 1 = Right side

(sec θ - tan θ)(sec θ + tan θ) = 1 ✓
Final answer:

The identity (sec θ - tan θ)(sec θ + tan θ) = 1 is verified.

Applied rules:

• Difference of squares: (a - b)(a + b) = a² - b²

• Derived identity: sec²θ - tan²θ = 1

• Algebraic expansion: Distribute terms correctly

5 Application problem
Exercise 5
If sin θ = 1/3 and θ is in Quadrant II, find the exact values of cos θ, tan θ, sec θ, csc θ, and cot θ using trigonometric identities.
Definition:

Quadrant signs: In Quadrant II: sine is positive, cosine is negative, tangent is negative.

Given
sin θ = 1/3, θ in QII
Find
cos θ, tan θ, sec θ, csc θ, cot θ
Identities
sin²+cos²=1, tan=sin/cos, etc.
Step 1: Find cos θ using Pythagorean identity

sin²θ + cos²θ = 1

(1/3)² + cos²θ = 1

1/9 + cos²θ = 1

cos²θ = 1 - 1/9 = 8/9

cos θ = ±√(8/9) = ±2√2/3

Step 2: Determine sign of cos θ

Since θ is in Quadrant II, cosine is negative

Therefore: cos θ = -2√2/3

Step 3: Find tan θ

tan θ = sin θ/cos θ = (1/3)/(-2√2/3) = (1/3) × (-3/(2√2)) = -1/(2√2)

Rationalize: tan θ = -1/(2√2) × √2/√2 = -√2/4

Step 4: Find reciprocal functions

sec θ = 1/cos θ = 1/(-2√2/3) = -3/(2√2) = -3√2/4

csc θ = 1/sin θ = 1/(1/3) = 3

cot θ = 1/tan θ = 1/(-√2/4) = -4/√2 = -2√2

cos θ = -2√2/3
tan θ = -√2/4
sec θ = -3√2/4
csc θ = 3
cot θ = -2√2
Final answer:

cos θ = -2√2/3, tan θ = -√2/4, sec θ = -3√2/4, csc θ = 3, cot θ = -2√2

Applied rules:

• Pythagorean identity: sin²θ + cos²θ = 1

• Quadrant signs: Determine correct signs

• Reciprocal identities: sec θ = 1/cos θ, etc.

• Rationalization: Remove radicals from denominators

Detailed Summary: Trig Identities Intro
sin²θ + cos²θ = 1
Pythagorean Identity
Key definitions:

Trigonometric identity: An equation involving trigonometric functions that is true for all values of the variable for which both sides are defined.

Pythagorean identities: sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, 1 + cot²θ = csc²θ

Reciprocal identities: sec θ = 1/cos θ, csc θ = 1/sin θ, cot θ = 1/tan θ

Identity Manipulation Methods:
  1. Substitution: Replace functions with their equivalents using identities
  2. Algebraic manipulation: Factor, expand, or simplify expressions
  3. Common denominators: Combine fractions when needed
  4. Converting to sine/cosine: Express everything in terms of sin and cos
  5. Working with both sides: Manipulate both sides independently toward common expression
Tip 1: Memorize the three Pythagorean identities and three reciprocal identities.
Tip 2: Always consider the quadrant when determining signs of trig functions.
Tip 3: When proving identities, start with the more complex side.
Tip 4: Check your work by substituting a specific angle to verify the identity holds.

Common errors: Forgetting signs in different quadrants, misapplying identities, algebraic mistakes in simplification.
Exam preparation: Practice deriving identities, memorize fundamental ones, master algebraic manipulations.
Essential identities to memorize:
Type Identity Name
Pythagorean sin²θ + cos²θ = 1 Fundamental
Pythagorean 1 + tan²θ = sec²θ Derived
Pythagorean 1 + cot²θ = csc²θ Derived
Reciprocal sec θ = 1/cos θ Secant
Reciprocal csc θ = 1/sin θ Cosecant
Reciprocal cot θ = 1/tan θ Cotangent
Trigonometric Identities: Visualization
Exercise 6: Identity Verification
Compare sin²θ + cos²θ and 1 for various values of θ

Analysis: The chart demonstrates that sin²θ + cos²θ always equals 1, verifying the fundamental Pythagorean identity.

  • sin²θ + cos²θ remains constant at 1 for all values of θ
  • This validates the Pythagorean identity across all quadrants
  • The identity holds regardless of the angle's size

Questions & Answers

Question: How do I remember all these trigonometric identities? There seem to be so many!

Answer: You don't need to memorize dozens of identities! Focus on the fundamentals:

The Big Three to Master:

  1. sin²θ + cos²θ = 1 (fundamental Pythagorean identity)
  2. tan θ = sin θ/cos θ (definition of tangent)
  3. sec θ = 1/cos θ, csc θ = 1/sin θ, cot θ = 1/tan θ (reciprocal identities)

Everything Else Derives From These:

  • Divide sin²θ + cos²θ = 1 by cos²θ to get: tan²θ + 1 = sec²θ
  • Divide sin²θ + cos²θ = 1 by sin²θ to get: 1 + cot²θ = csc²θ

Focus on understanding the derivations rather than rote memorization. Practice using the identities in problems to build familiarity!

Question: When proving identities, how do I know whether to start with the left side or the right side?

Answer: Generally, start with the more complex side of the equation. Here's the strategy:

Guidelines for choosing which side to start with:

  • More terms: If one side has more terms, start with that side
  • More operations: If one side involves more operations (multiplication, division, addition), start there
  • Complex fractions: If one side has complex fractions, start with that side
  • Multiple trig functions: If one side combines multiple trig functions, try starting there

Example: To prove (1 + tan²θ)(cos²θ) = 1, start with the left side since it's more complex.

Sometimes you might need to work on both sides independently until they meet in the middle!

Question: I'm confused about when to use the Pythagorean identity versus the reciprocal identity. How do I decide?

Answer: The choice depends on what functions appear in your expression:

Use Pythagorean identities when you see:

  • Squares of trig functions: sin²θ, cos²θ, tan²θ, etc.
  • Expressions like sin²θ + cos²θ or 1 - sin²θ
  • You need to convert between different squared functions

Use reciprocal identities when you see:

  • sec θ, csc θ, or cot θ that you want to convert to basic functions
  • You want to eliminate reciprocal functions
  • You have expressions like sec θ · cos θ (which equals 1)

Example: For sin²θ + cos²θ, use Pythagorean: sin²θ + cos²θ = 1

Example: For sec θ · cos θ, use reciprocal: sec θ · cos θ = (1/cos θ) · cos θ = 1

Look at the functions present and choose the identity that simplifies the expression!

Question: In exercise 5, why did we need to consider the quadrant? Why couldn't we just take the positive square root?

Answer: The quadrant is crucial because it determines the sign of trigonometric functions!

When we solved: cos²θ = 8/9

Taking the square root: cos θ = ±√(8/9) = ±2√2/3

This gives us two possible values, but only one is correct based on the quadrant.

Quadrant Rules:

  • Quadrant I: All functions positive
  • Quadrant II: Sine positive, cosine and tangent negative
  • Quadrant III: Tangent positive, sine and cosine negative
  • Quadrant IV: Cosine positive, sine and tangent negative

Since θ is in Quadrant II, cosine must be negative, so cos θ = -2√2/3.

Without quadrant information, we'd have to leave the answer as ±, but with it, we can determine the exact value.

Question: How can I check if I've made a mistake when working with trig identities?

Answer: Here are several verification techniques:

1. Substitution Check: Pick a specific angle (like 30° or 45°) and substitute into both sides of your identity to see if they're equal.

2. Known Values: Use special angles where you know the exact trig values (30°, 45°, 60°, 90°).

3. Work Backwards: If you derived a new identity, verify by working backwards to the original identity.

4. Check Your Steps: Review each algebraic step for common errors like sign mistakes or incorrect cancellations.

5. Use Reciprocal Relationships: If you found sin θ, check if 1/sin θ gives you csc θ.

Example: If you found sin θ = 1/3 and cos θ = -2√2/3, verify that (1/3)² + (-2√2/3)² = 1/9 + 8/9 = 1 ✓

Always take time to verify your answers, especially when working with complex expressions!