📐 Level 2 - Topic 9: Sum and Difference Identities - Expanding Trigonometric Toolkit ➕➖🔗

1) Expanding the Toolkit - Sum and Difference Identities ➕➖

Having established the fundamental trigonometric identities, we now expand our toolkit with sum and difference identities. These identities allow us to express trigonometric functions of sums or differences of angles in terms of trigonometric functions of the individual angles. They are immensely useful in simplifying expressions, solving equations, and in calculus and various applications.

What are Sum and Difference Identities?

  • They are formulas that express trigonometric functions of \( (A + B) \) and \( (A - B) \) in terms of trigonometric functions of \( A \) and \( B \) separately.
  • These identities are essential for breaking down complex angles into simpler ones and for combining trigonometric expressions.

Why are Sum and Difference Identities Important?

  • Simplifying Expressions: They help simplify trigonometric expressions involving sums or differences of angles.
  • Calculating Exact Values: They enable us to find exact trigonometric values for angles that are sums or differences of known angles (like \( 15^\circ = 45^\circ - 30^\circ \)).
  • Solving Equations: They are used to transform trigonometric equations into solvable forms.
  • Calculus and Advanced Math: Crucial in calculus, differential equations, and areas of physics and engineering.

In this topic, we will learn the sum and difference identities for sine, cosine, and tangent, understand how to use them for simplification, calculation, and verification, and solve problems using these powerful identities. Let's expand our trigonometric capabilities! ➕➖🔗

2) Sum and Difference Identities for Sine - Sine of Sum and Difference 🎶

Let's start with the sum and difference identities for the sine function.

Definition: Sum and Difference Identities for Sine

For any angles \( A \) and \( B \):

Sine of a Sum:

\( \sin(A + B) = \sin(A)\cos(B) + \cos(A)\sin(B) \)

Sine of a Difference:

\( \sin(A - B) = \sin(A)\cos(B) - \cos(A)\sin(B) \)

Notice the pattern:
  • For sine of a sum or difference, the identities involve both sine and cosine of individual angles.
  • For sine of a sum, it's a sum of products; for sine of a difference, it's a difference of products.
  • The operation sign in the angle \( (+\text{ or } -) \) is the same as the sign in the expanded form.

Example 1: Using Sine Sum Identity

Find the exact value of \( \sin(75^\circ) \) using sum identity.

  1. Express \( 75^\circ \) as a sum of known angles: \( 75^\circ = 45^\circ + 30^\circ \). So, \( A = 45^\circ, B = 30^\circ \).
  2. Apply sine sum identity: \( \sin(A + B) = \sin(A)\cos(B) + \cos(A)\sin(B) \).

    \( \sin(75^\circ) = \sin(45^\circ + 30^\circ) = \sin(45^\circ)\cos(30^\circ) + \cos(45^\circ)\sin(30^\circ) \)

  3. Substitute known values: \( \sin(45^\circ) = \cos(45^\circ) = \frac{\sqrt{2}}{2} \), \( \cos(30^\circ) = \frac{\sqrt{3}}{2} \), \( \sin(30^\circ) = \frac{1}{2} \).

    \( \sin(75^\circ) = \left(\frac{\sqrt{2}}{2}\right)\left(\frac{\sqrt{3}}{2}\right) + \left(\frac{\sqrt{2}}{2}\right)\left(\frac{1}{2}\right) \)

  4. Simplify:

    \( \sin(75^\circ) = \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} = \frac{\sqrt{6} + \sqrt{2}}{4} \)

Solution: \( \sin(75^\circ) = \frac{\sqrt{6} + \sqrt{2}}{4} \).

Example 2: Using Sine Difference Identity

Find the exact value of \( \sin(15^\circ) \) using difference identity.

  1. Express \( 15^\circ \) as a difference of known angles: \( 15^\circ = 45^\circ - 30^\circ \). So, \( A = 45^\circ, B = 30^\circ \).
  2. Apply sine difference identity: \( \sin(A - B) = \sin(A)\cos(B) - \cos(A)\sin(B) \).

    \( \sin(15^\circ) = \sin(45^\circ - 30^\circ) = \sin(45^\circ)\cos(30^\circ) - \cos(45^\circ)\sin(30^\circ) \)

  3. Substitute known values: \( \sin(45^\circ) = \cos(45^\circ) = \frac{\sqrt{2}}{2} \), \( \cos(30^\circ) = \frac{\sqrt{3}}{2} \), \( \sin(30^\circ) = \frac{1}{2} \).

    \( \sin(15^\circ) = \left(\frac{\sqrt{2}}{2}\right)\left(\frac{\sqrt{3}}{2}\right) - \left(\frac{\sqrt{2}}{2}\right)\left(\frac{1}{2}\right) \)

  4. Simplify:

    \( \sin(15^\circ) = \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} = \frac{\sqrt{6} - \sqrt{2}}{4} \)

Solution: \( \sin(15^\circ) = \frac{\sqrt{6} - \sqrt{2}}{4} \).

3) Sum and Difference Identities for Cosine - Cosine of Sum and Difference 🎵

Now, let's explore the sum and difference identities for the cosine function.

Definition: Sum and Difference Identities for Cosine

For any angles \( A \) and \( B \):

Cosine of a Sum:

\( \cos(A + B) = \cos(A)\cos(B) - \sin(A)\sin(B) \)

Cosine of a Difference:

\( \cos(A - B) = \cos(A)\cos(B) + \sin(A)\sin(B) \)

Notice the pattern:
  • For cosine of a sum or difference, the identities involve only cosine terms and only sine terms multiplied together.
  • For cosine of a sum, it's a difference of products; for cosine of a difference, it's a sum of products.
  • The operation sign in the angle \( (+\text{ or } -) \) is opposite to the sign in the expanded form.

Example 3: Using Cosine Sum Identity

Find the exact value of \( \cos(75^\circ) \) using sum identity.

  1. Express \( 75^\circ \) as a sum: \( 75^\circ = 45^\circ + 30^\circ \). So, \( A = 45^\circ, B = 30^\circ \).
  2. Apply cosine sum identity: \( \cos(A + B) = \cos(A)\cos(B) - \sin(A)\sin(B) \).

    \( \cos(75^\circ) = \cos(45^\circ + 30^\circ) = \cos(45^\circ)\cos(30^\circ) - \sin(45^\circ)\sin(30^\circ) \)

  3. Substitute known values: \( \sin(45^\circ) = \cos(45^\circ) = \frac{\sqrt{2}}{2} \), \( \cos(30^\circ) = \frac{\sqrt{3}}{2} \), \( \sin(30^\circ) = \frac{1}{2} \).

    \( \cos(75^\circ) = \left(\frac{\sqrt{2}}{2}\right)\left(\frac{\sqrt{3}}{2}\right) - \left(\frac{\sqrt{2}}{2}\right)\left(\frac{1}{2}\right) \)

  4. Simplify:

    \( \cos(75^\circ) = \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} = \frac{\sqrt{6} - \sqrt{2}}{4} \)

Solution: \( \cos(75^\circ) = \frac{\sqrt{6} - \sqrt{2}}{4} \).

Example 4: Using Cosine Difference Identity

Find the exact value of \( \cos(15^\circ) \) using difference identity.

  1. Express \( 15^\circ \) as a difference: \( 15^\circ = 45^\circ - 30^\circ \). So, \( A = 45^\circ, B = 30^\circ \).
  2. Apply cosine difference identity: \( \cos(A - B) = \cos(A)\cos(B) + \sin(A)\sin(B) \).

    \( \cos(15^\circ) = \cos(45^\circ - 30^\circ) = \cos(45^\circ)\cos(30^\circ) + \sin(45^\circ)\sin(30^\circ) \)

  3. Substitute known values: \( \sin(45^\circ) = \cos(45^\circ) = \frac{\sqrt{2}}{2} \), \( \cos(30^\circ) = \frac{\sqrt{3}}{2} \), \( \sin(30^\circ) = \frac{1}{2} \).

    \( \cos(15^\circ) = \left(\frac{\sqrt{2}}{2}\right)\left(\frac{\sqrt{3}}{2}\right) + \left(\frac{\sqrt{2}}{2}\right)\left(\frac{1}{2}\right) \)

  4. Simplify:

    \( \cos(15^\circ) = \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} = \frac{\sqrt{6} + \sqrt{2}}{4} \)

Solution: \( \cos(15^\circ) = \frac{\sqrt{6} + \sqrt{2}}{4} \).

4) Sum and Difference Identities for Tangent - Tangent of Sum and Difference 🥁

Finally, let's look at the sum and difference identities for the tangent function.

Definition: Sum and Difference Identities for Tangent

For any angles \( A \) and \( B \), provided \( \cos(A+B) \neq 0 \) and \( \cos(A-B) \neq 0 \):

Tangent of a Sum:

\( \tan(A + B) = \frac{\tan(A) + \tan(B)}{1 - \tan(A)\tan(B)} \)

Tangent of a Difference:

\( \tan(A - B) = \frac{\tan(A) - \tan(B)}{1 + \tan(A)\tan(B)} \)

Notice the pattern:
  • For tangent of a sum, the numerator is the sum of tangents, and the denominator is 1 minus the product of tangents.
  • For tangent of a difference, the numerator is the difference of tangents, and the denominator is 1 plus the product of tangents.
  • The sign in the numerator matches the sign in the angle \( (+\text{ or } -) \), while the sign in the denominator is opposite.

Example 5: Using Tangent Sum Identity

Find the exact value of \( \tan(105^\circ) \) using sum identity.

  1. Express \( 105^\circ \) as a sum: \( 105^\circ = 60^\circ + 45^\circ \). So, \( A = 60^\circ, B = 45^\circ \).
  2. Apply tangent sum identity: \( \tan(A + B) = \frac{\tan(A) + \tan(B)}{1 - \tan(A)\tan(B)} \).

    \( \tan(105^\circ) = \tan(60^\circ + 45^\circ) = \frac{\tan(60^\circ) + \tan(45^\circ)}{1 - \tan(60^\circ)\tan(45^\circ)} \)

  3. Substitute known values: \( \tan(60^\circ) = \sqrt{3} \), \( \tan(45^\circ) = 1 \).

    \( \tan(105^\circ) = \frac{\sqrt{3} + 1}{1 - (\sqrt{3})(1)} = \frac{\sqrt{3} + 1}{1 - \sqrt{3}} \)

  4. Rationalize denominator (optional, but common for simplified form): Multiply numerator and denominator by the conjugate of the denominator \( (1 + \sqrt{3}) \).

    \( \tan(105^\circ) = \frac{(\sqrt{3} + 1)}{(1 - \sqrt{3})} \cdot \frac{(1 + \sqrt{3})}{(1 + \sqrt{3})} = \frac{(\sqrt{3} + 1)^2}{1^2 - (\sqrt{3})^2} = \frac{3 + 2\sqrt{3} + 1}{1 - 3} = \frac{4 + 2\sqrt{3}}{-2} = -2 - \sqrt{3} \)

Solution: \( \tan(105^\circ) = -2 - \sqrt{3} \).

Example 6: Using Tangent Difference Identity

Find the exact value of \( \tan(15^\circ) \) using difference identity.

  1. Express \( 15^\circ \) as a difference: \( 15^\circ = 45^\circ - 30^\circ \). So, \( A = 45^\circ, B = 30^\circ \).
  2. Apply tangent difference identity: \( \tan(A - B) = \frac{\tan(A) - \tan(B)}{1 + \tan(A)\tan(B)} \).

    \( \tan(15^\circ) = \tan(45^\circ - 30^\circ) = \frac{\tan(45^\circ) - \tan(30^\circ)}{1 + \tan(45^\circ)\tan(30^\circ)} \)

  3. Substitute known values: \( \tan(45^\circ) = 1 \), \( \tan(30^\circ) = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3} \).

    \( \tan(15^\circ) = \frac{1 - \frac{\sqrt{3}}{3}}{1 + (1)\left(\frac{\sqrt{3}}{3}\right)} = \frac{\frac{3 - \sqrt{3}}{3}}{\frac{3 + \sqrt{3}}{3}} = \frac{3 - \sqrt{3}}{3 + \sqrt{3}} \)

  4. Rationalize denominator (optional simplification): Multiply numerator and denominator by the conjugate of the denominator \( (3 - \sqrt{3}) \).

    \( \tan(15^\circ) = \frac{(3 - \sqrt{3})}{(3 + \sqrt{3})} \cdot \frac{(3 - \sqrt{3})}{(3 - \sqrt{3})} = \frac{(3 - \sqrt{3})^2}{3^2 - (\sqrt{3})^2} = \frac{9 - 6\sqrt{3} + 3}{9 - 3} = \frac{12 - 6\sqrt{3}}{6} = 2 - \sqrt{3} \)

Solution: \( \tan(15^\circ) = 2 - \sqrt{3} \).

5) Verifying Identities using Sum and Difference Identities 🚀

Example 7: Verifying Identity using Sum Identity for Cosine

Verify the identity: \( \cos(\frac{\pi}{2} - \theta) = \sin(\theta) \).

  1. Start with the left side (LS): \( \text{LS} = \cos(\frac{\pi}{2} - \theta) \).
  2. Apply cosine difference identity: \( \cos(A - B) = \cos(A)\cos(B) + \sin(A)\sin(B) \) with \( A = \frac{\pi}{2}, B = \theta \).

    \( \cos(\frac{\pi}{2} - \theta) = \cos(\frac{\pi}{2})\cos(\theta) + \sin(\frac{\pi}{2})\sin(\theta) \)

  3. Substitute known values: \( \cos(\frac{\pi}{2}) = 0 \), \( \sin(\frac{\pi}{2}) = 1 \).

    \( \cos(\frac{\pi}{2} - \theta) = (0)\cos(\theta) + (1)\sin(\theta) \)

  4. Simplify: \( \text{LS} = 0 + \sin(\theta) = \sin(\theta) \).
  5. Compare with the right side (RS): \( \text{RS} = \sin(\theta) \).
  6. Conclusion: Since \( \text{LS} = \text{RS} \), the identity \( \cos(\frac{\pi}{2} - \theta) = \sin(\theta) \) is verified. (This is a cofunction identity, now derived from sum/difference).

Example 8: Verifying Identity using Sum Identity for Sine

Verify the identity: \( \sin(\pi + \theta) = -\sin(\theta) \).

  1. Start with the left side (LS): \( \text{LS} = \sin(\pi + \theta) \).
  2. Apply sine sum identity: \( \sin(A + B) = \sin(A)\cos(B) + \cos(A)\sin(B) \) with \( A = \pi, B = \theta \).

    \( \sin(\pi + \theta) = \sin(\pi)\cos(\theta) + \cos(\pi)\sin(\theta) \)

  3. Substitute known values: \( \sin(\pi) = 0 \), \( \cos(\pi) = -1 \).

    \( \sin(\pi + \theta) = (0)\cos(\theta) + (-1)\sin(\theta) \)

  4. Simplify: \( \text{LS} = 0 - \sin(\theta) = -\sin(\theta) \).
  5. Compare with the right side (RS): \( \text{RS} = -\sin(\theta) \).
  6. Conclusion: Since \( \text{LS} = \text{RS} \), the identity \( \sin(\pi + \theta) = -\sin(\theta) \) is verified. (This is a reduction formula, now derived from sum identity).

6) Solving Equations using Sum and Difference Identities 🚀

Example 9: Solving Equation using Cosine Difference Identity

Solve the equation \( \cos(2x)\cos(x) + \sin(2x)\sin(x) = \frac{\sqrt{3}}{2} \) for principal solutions in \( [0, 2\pi) \).

  1. Recognize the cosine difference pattern: The left side is in the form \( \cos(A)\cos(B) + \sin(A)\sin(B) \) which is \( \cos(A - B) \) with \( A = 2x \) and \( B = x \).
  2. Apply cosine difference identity: Rewrite the equation as \( \cos(2x - x) = \frac{\sqrt{3}}{2} \Rightarrow \cos(x) = \frac{\sqrt{3}}{2} \).
  3. Solve the basic cosine equation \( \cos(x) = \frac{\sqrt{3}}{2} \): Principal value \( \arccos(\frac{\sqrt{3}}{2}) = \frac{\pi}{6} \). Principal solutions in \( [0, 2\pi) \) are \( x = \frac{\pi}{6} \) and \( x = 2\pi - \frac{\pi}{6} = \frac{11\pi}{6} \).

Solution: Principal solutions are \( x = \frac{\pi}{6}, \frac{11\pi}{6} \).

Example 10: Solving Equation using Sine Sum Identity

Solve the equation \( \sin(x)\cos(\frac{\pi}{3}) + \cos(x)\sin(\frac{\pi}{3}) = 1 \) for principal solutions in \( [0, 2\pi) \).

  1. Recognize the sine sum pattern: The left side is in the form \( \sin(A)\cos(B) + \cos(A)\sin(B) \) which is \( \sin(A + B) \) with \( A = x \) and \( B = \frac{\pi}{3} \).
  2. Apply sine sum identity: Rewrite the equation as \( \sin(x + \frac{\pi}{3}) = 1 \).
  3. Solve \( \sin(y) = 1 \) where \( y = x + \frac{\pi}{3} \): Principal solution for \( \sin(y) = 1 \) is \( y = \frac{\pi}{2} \). General solution for \( y \) is \( y = \frac{\pi}{2} + 2n\pi \).
  4. Solve for \( x \): \( x + \frac{\pi}{3} = \frac{\pi}{2} + 2n\pi \Rightarrow x = \frac{\pi}{2} - \frac{\pi}{3} + 2n\pi = \frac{3\pi - 2\pi}{6} + 2n\pi = \frac{\pi}{6} + 2n\pi \).
  5. Principal solutions for \( x \) in \( [0, 2\pi) \): For \( n = 0 \), \( x = \frac{\pi}{6} \). For \( n = 1 \), \( x = \frac{\pi}{6} + 2\pi > 2\pi \). For \( n = -1 \), \( x < 0 \). So, the only principal solution is \( x = \frac{\pi}{6} \).

Solution: Principal solution is \( x = \frac{\pi}{6} \).

7) Practice Questions 🎯

7.1 Fundamental – Using Sum and Difference Identities for Calculation and Verification

1. Find the exact value of \( \sin(105^\circ) \).

2. Find the exact value of \( \cos(105^\circ) \).

3. Find the exact value of \( \tan(75^\circ) \).

4. Find the exact value of \( \sin(285^\circ) \) (Hint: \( 285^\circ = 360^\circ - 75^\circ \) or \( 285^\circ = 225^\circ + 60^\circ \)).

5. Verify the identity: \( \sin(\frac{\pi}{2} + \theta) = \cos(\theta) \).

6. Verify the identity: \( \cos(\pi + \theta) = -\cos(\theta) \).

7. Verify the identity: \( \tan(\pi - \theta) = -\tan(\theta) \).

8. Verify the identity: \( \sin(2\theta) = \sin(\theta + \theta) = 2\sin(\theta)\cos(\theta) \) (Start of Double Angle Identity derivation).

9. Verify the identity: \( \cos(2\theta) = \cos(\theta + \theta) = \cos^2(\theta) - \sin^2(\theta) \) (Start of Double Angle Identity derivation).

10. Simplify: \( \sin(3x)\cos(2x) + \cos(3x)\sin(2x) \).

11. Simplify: \( \cos(5x)\cos(x) - \sin(5x)\sin(x) \).

12. Simplify: \( \frac{\tan(4x) - \tan(x)}{1 + \tan(4x)\tan(x)} \).

7.2 Challenging – Advanced Verification and Equation Solving 💪🚀

1. Verify: \( \frac{\sin(A + B)}{\sin(A - B)} = \frac{\tan(A) + \tan(B)}{\tan(A) - \tan(B)} \).

2. Verify: \( \tan(x + \frac{\pi}{4}) = \frac{1 + \tan(x)}{1 - \tan(x)} \).

3. Verify: \( \cos(3x) = \cos(2x + x) = 4\cos^3(x) - 3\cos(x) \) (Use cosine sum and double angle identities).

4. Solve the equation: \( \sin(x + \frac{\pi}{4}) + \sin(x - \frac{\pi}{4}) = 1 \) for principal solutions in \( [0, 2\pi) \).

5. Solve the equation: \( \cos(x + 60^\circ) - \cos(x) = 1 \) for principal solutions in \( [0^\circ, 360^\circ) \).

6. (Conceptual) Explain why \( \sin(A+B) \neq \sin(A) + \sin(B) \) in general. Illustrate with a numerical example.

7. (Challenging Verification) Verify: \( \frac{\cos(A + B) + \cos(A - B)}{\sin(A + B) - \sin(A - B)} = \cot(B) \).

8) Summary - Sum and Difference Identities - Expanding Horizons 🎉

  • Sum and Difference Identities: Formulas for \( \sin(A \pm B) \), \( \cos(A \pm B) \), \( \tan(A \pm B) \) in terms of \( \sin(A) \), \( \cos(A) \), \( \tan(A) \), \( \sin(B) \), \( \cos(B) \), \( \tan(B) \).
  • Sine Identities: \( \sin(A \pm B) = \sin(A)\cos(B) \pm \cos(A)\sin(B) \) (sign matches).
  • Cosine Identities: \( \cos(A \pm B) = \cos(A)\cos(B) \mp \sin(A)\sin(B) \) (sign opposite).
  • Tangent Identities: \( \tan(A \pm B) = \frac{\tan(A) \pm \tan(B)}{1 \mp \tan(A)\tan(B)} \) (sign in numerator matches, denominator opposite).
  • Applications: Calculating exact values, simplifying expressions, verifying identities, solving equations.
  • Techniques: Express angles as sums or differences of known angles, recognize patterns in equations, apply identities strategically, algebraic simplification.

Fantastic! You've now mastered the sum and difference identities, adding a powerful set of tools to your trigonometric arsenal. These identities not only enable you to calculate exact trigonometric values for a wider range of angles but also equip you to tackle more complex trigonometric expressions and equations. As you move towards more advanced topics, you'll find these identities indispensable. Keep practicing, and you'll skillfully wield these identities to unlock even more of the trigonometric world! ➕➖🔗📐🌟

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