Trigonometric Identities

🔄 Quick Recap

In previous sections, we learned about the six trigonometric ratios and their values for specific angles. Now we'll explore special relationships between these ratios called "trigonometric identities." These are equations that are always true for any angle (where the expressions are defined).

📚 What is a Trigonometric Identity?

A trigonometric identity is an equation involving trigonometric ratios that is true for all values of the angles involved (as long as the expressions are defined). These identities are extremely useful because they help us:

• Simplify complex trigonometric expressions
• Convert one trigonometric ratio to another
• Prove other mathematical relationships
• Solve problems more efficiently

🖼️ The Fundamental Trigonometric Identity

The most important trigonometric identity is:

sin² A + cos² A = 1

Let's see how we can prove this using the Pythagorean theorem!

Understanding the Visual Proof

The diagram above shows:

• A unit circle (a circle with radius 1) centered at the origin O
• A point P on the circle, connected to O by a green line (the radius)
• A right triangle formed by:
  • The radius OP (which equals 1)
  • The horizontal distance from O to the vertical line through P (labeled "cos θ")
  • The vertical distance from that point to P (labeled "sin θ")

This creates a right triangle where:

• The hypotenuse is 1 (the radius)
• The base is cos θ
• The height is sin θ

According to the Pythagorean theorem, in any right triangle:

(base)² + (height)² = (hypotenuse)²

Substituting our values:

(cos θ)² + (sin θ)² = 1²
cos² θ + sin² θ = 1

And that's our fundamental identity! This relationship works for any angle θ, not just the 45° angle shown in the diagram.

Alternative Proof Using a Right Triangle

We can also prove this identity using any right triangle ABC, right-angled at B:

By the Pythagorean theorem:

AB² + BC² = AC²

Dividing both sides by AC²:

(AB/AC)² + (BC/AC)² = (AC/AC)²
(cos A)² + (sin A)² = 1
sin² A + cos² A = 1

📚 Other Important Identities

From this fundamental identity, we can derive two other important identities.

1. The Tangent-Secant Identity

If we divide both sides of sin² A + cos² A = 1 by cos² A, we get:

sin² A/cos² A + cos² A/cos² A = 1/cos² A

Let's simplify each term:

• sin² A/cos² A = (sin A/cos A)² = tan² A
• cos² A/cos² A = 1
• 1/cos² A = sec² A

So our equation becomes:

tan² A + 1 = sec² A

This makes sense when you think about it: as the angle gets closer to 90°, both tan A and sec A get very large, approaching infinity.

2. The Cotangent-Cosecant Identity

Similarly, if we divide both sides of sin² A + cos² A = 1 by sin² A, we get:

sin² A/sin² A + cos² A/sin² A = 1/sin² A

Let's simplify:

• sin² A/sin² A = 1
• cos² A/sin² A = (cos A/sin A)² = cot² A
• 1/sin² A = cosec² A

Our equation becomes:

1 + cot² A = cosec² A

⚖️ Quick Summary of Key Identities

1. Pythagorean Identities (derived from the Pythagorean theorem):

   sin² A + cos² A = 1
   tan² A + 1 = sec² A
   1 + cot² A = cosec² A

2. Quotient Identities (relating ratios to each other):

   tan A = sin A / cos A
   cot A = cos A / sin A

3. Reciprocal Identities (defining the secondary ratios):

   cosec A = 1 / sin A
   sec A = 1 / cos A
   cot A = 1 / tan A

These identities form the foundation of trigonometry and will help you solve many kinds of problems!

📚 Using Identities to Find Unknown Trigonometric Ratios

One of the most practical uses of trigonometric identities is finding all trigonometric ratios when you know just one. Let's see how this works with an example.

Example:

If sin A = 3/5, find all other trigonometric ratios.

Solution: We know sin A = 3/5.

Step 1: Find cos A using the Pythagorean identity sin² A + cos² A = 1

cos² A = 1 - sin² A 
cos² A = 1 - (3/5)² 
cos² A = 1 - 9/25 
cos² A = 16/25

Since angle A is acute (we can tell because sin A is positive and less than 1), cos A is also positive.

cos A = √(16/25) = 4/5

Step 2: Find the remaining ratios using the relationships between them:

tan A = sin A / cos A = (3/5) ÷ (4/5) = 3/4

cosec A = 1 / sin A = 1 ÷ (3/5) = 5/3

sec A = 1 / cos A = 1 ÷ (4/5) = 5/4

cot A = 1 / tan A = 1 ÷ (3/4) = 4/3

So all six trigonometric ratios for this angle are:

• sin A = 3/5
• cos A = 4/5
• tan A = 3/4
• cosec A = 5/3
• sec A = 5/4
• cot A = 4/3

Let's verify that these values satisfy our identities:

sin² A + cos² A = (3/5)² + (4/5)² = 9/25 + 16/25 = 25/25 = 1 ✓

tan² A + 1 = (3/4)² + 1 = 9/16 + 16/16 = 9/16 + 16/16 = 25/16
sec² A = (5/4)² = 25/16 ✓

1 + cot² A = 1 + (4/3)² = 1 + 16/9 = 9/9 + 16/9 = 25/9
cosec² A = (5/3)² = 25/9 ✓

All identities check out! This confirms that our calculations are correct.

✅ Solved Examples

Let's practice using these identities by proving some more complex relationships.

Example 1:

Prove that (1 - cos A) / sin A + sin A / (1 - cos A) = 2 cosec A

Solution: Let's simplify the left-hand side (LHS) step by step:

Step 1: Find a common denominator for the two fractions

LHS = (1 - cos A) / sin A + sin A / (1 - cos A)
    = ((1 - cos A)² + sin² A) / (sin A × (1 - cos A))

Step 2: Simplify the numerator

(1 - cos A)² + sin² A = 1 - 2cos A + cos² A + sin² A

Step 3: Use the identity sin² A + cos² A = 1

1 - 2cos A + cos² A + sin² A = 1 - 2cos A + cos² A + (1 - cos² A)
                             = 1 - 2cos A + cos² A + 1 - cos² A
                             = 2 - 2cos A
                             = 2(1 - cos A)

Step 4: Substitute back into our expression

LHS = 2(1 - cos A) / (sin A × (1 - cos A))
    = 2 / sin A
    = 2 cosec A

Therefore, LHS = RHS, and the identity is proved.

Example 2:

Prove that (sec A - tan A) × (sec A + tan A) = 1

Solution: This is an example of the difference of squares formula (a - b)(a + b) = a² - b².

Step 1: Apply the difference of squares formula

LHS = (sec A - tan A) × (sec A + tan A)
    = sec² A - tan² A

Step 2: Use the identity tan² A + 1 = sec² A, which we can rearrange to sec² A - tan² A = 1

LHS = sec² A - tan² A
    = 1

Therefore, LHS = RHS, and the identity is proved.

This identity has a beautiful geometric interpretation: in a right triangle, if we construct a certain tangent line to the unit circle, the product of two related segments equals 1.

🧮 Mathematical Corner: Express All Ratios in Terms of One

Another powerful application of identities is expressing all trigonometric ratios in terms of just one. This is especially useful in calculus and other advanced mathematics.

If we know sin A = p (where p is some value between 0 and 1 for acute angles), we can find all other ratios:

Step 1: Find cos A using sin² A + cos² A = 1

cos² A = 1 - sin² A = 1 - p²

Since we're dealing with acute angles (0° < A < 90°), cos A is positive:

cos A = √(1 - p²)

Step 2: Find the remaining ratios:

tan A = sin A / cos A = p / √(1 - p²)

cot A = cos A / sin A = √(1 - p²) / p

sec A = 1 / cos A = 1 / √(1 - p²)

cosec A = 1 / sin A = 1 / p

Similarly, if we know any one of the six trigonometric ratios, we can find all others using identities.

⚠️ Common Misconceptions

1. Misconception: sin² A means sin(sin A). Fact: sin² A is shorthand for (sin A)², meaning the sine of angle A squared. It does not mean applying the sine function twice.

2. Misconception: Trigonometric identities work only for specific angles like 30°, 45°, or 60°. Fact: Trigonometric identities are true for all angles where the expressions are defined. We can use specific angles to verify them, but they work universally.

3. Misconception: The identity sin² A + cos² A = 1 works only for angles in right triangles. Fact: This identity is true for all angles, even those greater than 90°. The unit circle definition of trigonometric functions extends them beyond right triangles.

4. Misconception: If an equation is true for one angle, it's true for all angles. Fact: While genuine identities are true for all angles, some equations might only be true for specific angle values. We need to check that an equation works generally before calling it an identity.

🌍 Real-Life Applications

Trigonometric identities aren't just mathematical curiosities—they're used in many practical fields:

• Physics: When analyzing waves (like sound or light), identities help simplify complex equations for wave interference and resonance.

• Engineering: Electrical engineers use trigonometric identities when working with alternating currents and designing filters and circuits.

• Computer Graphics: When rotating objects in 3D space, trigonometric identities help simplify the calculations needed for realistic animations.

• Signal Processing: In audio processing and telecommunications, engineers use identities to analyze and manipulate signals efficiently.

• Navigation: GPS systems and aircraft navigation rely on trigonometric calculations that can be simplified using identities.

For example, in physics, when combining two waves with the same frequency but different phases, we use the identity:

sin A + sin B = 2 sin((A+B)/2) cos((A-B)/2)

This transforms a sum of sines into a product, making the mathematics much more manageable.

🧠 Memory Tricks

To remember the three Pythagorean identities:

1. Start with sin² A + cos² A = 1, which you can remember by thinking about a right triangle in a unit circle.

2. To get the second identity, think "divide by cos²" which gives tan² A + 1 = sec² A.

3. To get the third identity, think "divide by sin²" which gives 1 + cot² A = cosec² A.

Notice the pattern? In the first identity, the sin term comes first. In the second identity (after dividing by cos²), the tan term comes first. In the third identity (after dividing by sin²), the constant term comes first.

Another way to remember: the three identities relate to the three pairs of trigonometric functions:

• sin and cos (sin² + cos² = 1)
• tan and sec (tan² + 1 = sec²)
• cot and cosec (1 + cot² = cosec²)

🤔 Think About It!

1. Why do you think the identity sin² A + cos² A = 1 is related to the Pythagorean theorem? What is the geometric meaning of this relationship?

2. What would happen to our trigonometric identities if we were working on a sphere instead of a flat plane? (This question leads to spherical trigonometry, which is used in navigation and astronomy!)

3. Can you verify the identity sin² A + cos² A = 1 for A = 30°, 45°, and 60° using the values we learned in the previous section?

4. Why is it useful to be able to express all trigonometric ratios in terms of just one? When might this come in handy?

5. If you know the value of sin A, can you always determine the exact angle A? Why or why not?

🧪 Activity Time!

Verify an Identity

Let's verify the identity sin² A + cos² A = 1 for different angles:

1. For A = 30°:

   • sin 30° = 1/2
   • cos 30° = √3/2
   • sin² 30° + cos² 30° = (1/2)² + (√3/2)² = 1/4 + 3/4 = 1 ✓

2. For A = 45°:

   • sin 45° = 1/√2 = √2/2
   • cos 45° = 1/√2 = √2/2
   • sin² 45° + cos² 45° = (√2/2)² + (√2/2)² = 2/4 + 2/4 = 1 ✓

3. For A = 60°:

   • sin 60° = √3/2
   • cos 60° = 1/2
   • sin² 60° + cos² 60° = (√3/2)² + (1/2)² = 3/4 + 1/4 = 1 ✓

Try it yourself for other angles!

🔜 What Next?

In the next section, we'll solve more complex problems using the trigonometric ratios and identities we've learned. We'll see how these concepts can be applied to real-world situations like finding heights, distances, and angles in practical scenarios. The identities we've learned will help us simplify complex calculations and solve problems more efficiently!