Sum/difference and double-angle formulas

LVL: FREE

MODULE: Trigonometric Functions and Identities

[EXEC: MICRO_CORE]

✖️ 1. Applying sum and difference formulas for sine, cosine, and tangent explicitly

➕ Sum and Difference Formulas

  • Sine sum: sin⁡(A+B)=sin⁡Acos⁡B+cos⁡Asin⁡B\sin(A + B) = \sin A \cos B + \cos A \sin B
  • Sine difference: sin⁡(A−B)=sin⁡Acos⁡B−cos⁡Asin⁡B\sin(A - B) = \sin A \cos B - \cos A \sin B
  • Cosine sum: cos⁡(A+B)=cos⁡Acos⁡B−sin⁡Asin⁡B\cos(A + B) = \cos A \cos B - \sin A \sin B
  • Cosine difference: cos⁡(A−B)=cos⁡Acos⁡B+sin⁡Asin⁡B\cos(A - B) = \cos A \cos B + \sin A \sin B
  • Tangent sum: tan⁡(A+B)=tan⁡A+tan⁡B1−tan⁡Atan⁡B\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}
  • Tangent difference: tan⁡(A−B)=tan⁡A−tan⁡B1+tan⁡Atan⁡B\tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}

Example: sin⁡(45∘+30∘)=sin⁡45∘cos⁡30∘+cos⁡45∘sin⁡30∘=22⋅32+22⋅12=6+24\sin(45^\circ + 30^\circ) = \sin 45^\circ \cos 30^\circ + \cos 45^\circ \sin 30^\circ = \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2} = \frac{\sqrt{6} + \sqrt{2}}{4}

💡 Sine adds products, cosine flips the middle sign!

[EXEC: DEEP_COMPUTE]

1. Applying sum and difference formulas for sine, cosine, and tangent explicitly

Sum and Difference Formulas

The sum and difference formulas express trigonometric functions of angle sums or differences in terms of products of functions of individual angles. These formulas are fundamental identities: sin⁡(A±B)=sin⁡Acos⁡B±cos⁡Asin⁡B\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B, cos⁡(A±B)=cos⁡Acos⁡B∓sin⁡Asin⁡B\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B, and tan⁡(A±B)=tan⁡A±tan⁡B1∓tan⁡Atan⁡B\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}.

These formulas allow decomposition of complex angles into simpler components whose values are known.

Core Rules:

  • For sine: addition uses plus, subtraction uses minus in the second term
  • For cosine: addition uses minus, subtraction uses plus (opposite pattern)
  • For tangent: denominator sign is opposite to numerator sign
  • All formulas require both angles to be in the domain of the respective functions

These identities enable exact computation of trigonometric values and simplification of expressions involving angle combinations.

Example: sin⁡(45∘+30∘)=sin⁡45∘cos⁡30∘+cos⁡45∘sin⁡30∘=22⋅32+22⋅12=6+24\sin(45^\circ + 30^\circ) = \sin 45^\circ \cos 30^\circ + \cos 45^\circ \sin 30^\circ = \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2} = \frac{\sqrt{6} + \sqrt{2}}{4}

TASK_1[0 / 3]
LVL_2
EXEC: FORMULA

Write the exact algebraic expansion for cos⁡(x−y)\cos(x - y) using the difference formula. Use the variables xx and yy.

DEEP_COMPUTE
ULTRA
[EXEC: MICRO_CORE]

✖️ 2. Deriving and applying double-angle formulas

🔄 Double-Angle Formulas

  • Sine double: sin⁡2θ=2sin⁡θcos⁡θ\sin 2\theta = 2\sin\theta \cos\theta
  • Cosine double (3 forms): cos⁡2θ=cos⁡2θ−sin⁡2θ=2cos⁡2θ−1=1−2sin⁡2θ\cos 2\theta = \cos^2\theta - \sin^2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta
  • Tangent double: tan⁡2θ=2tan⁡θ1−tan⁡2θ\tan 2\theta = \frac{2\tan\theta}{1 - \tan^2\theta}
  • Derive by setting A=B=θA = B = \theta in sum formulas.
  • Choose the cosine form based on what you know (sine or cosine).

Example: If sin⁡θ=35\sin\theta = \frac{3}{5} and cos⁡θ=45\cos\theta = \frac{4}{5}, then sin⁡2θ=2⋅35⋅45=2425\sin 2\theta = 2 \cdot \frac{3}{5} \cdot \frac{4}{5} = \frac{24}{25}

💡 Double the angle, double the sine-cosine product!

[EXEC: DEEP_COMPUTE]

2. Deriving and applying double-angle formulas

Double-Angle Formulas

Double-angle formulas express trigonometric functions of 2θ2\theta in terms of functions of θ\theta. They are derived by setting A=B=θA = B = \theta in the sum formulas: sin⁡2θ=2sin⁡θcos⁡θ\sin 2\theta = 2\sin\theta\cos\theta, cos⁡2θ=cos⁡2θ−sin⁡2θ\cos 2\theta = \cos^2\theta - \sin^2\theta, and tan⁡2θ=2tan⁡θ1−tan⁡2θ\tan 2\theta = \frac{2\tan\theta}{1 - \tan^2\theta}.

These formulas reduce the argument of a trigonometric function by half, enabling simplification and exact value computation.

Core Rules:

  • sin⁡2θ\sin 2\theta has exactly one alternative form: 2sin⁡θcos⁡θ2\sin\theta\cos\theta
  • cos⁡2θ\cos 2\theta has three equivalent forms: cos⁡2θ−sin⁡2θ=2cos⁡2θ−1=1−2sin⁡2θ\cos^2\theta - \sin^2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta
  • tan⁡2θ\tan 2\theta is undefined when tan⁡2θ=1\tan^2\theta = 1 (i.e., θ=45∘+90∘k\theta = 45^\circ + 90^\circ k)
  • The cosine variants are derived using sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1

These formulas are essential for integration, solving equations, and analyzing periodic phenomena.

Example: cos⁡60∘=cos⁡(2⋅30∘)=1−2sin⁡230∘=1−2(12)2=12\cos 60^\circ = \cos(2 \cdot 30^\circ) = 1 - 2\sin^2 30^\circ = 1 - 2(\frac{1}{2})^2 = \frac{1}{2}

TASK_1[0 / 3]
LVL_2
EXEC: FORMULA

If sin⁡θ=0.6\sin \theta = 0.6, what is the exact numerical value of cos⁡2θ\cos 2\theta?

DEEP_COMPUTE
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[EXEC: MICRO_CORE]

✖️ 3. Finding exact values for non-standard angles using addition/subtraction

🎯 Exact Values for Non-Standard Angles

  • Break unusual angles into sums or differences of standard angles (30°, 45°, 60°).
  • Example angles: 15∘=45∘−30∘15^\circ = 45^\circ - 30^\circ, 75∘=45∘+30∘75^\circ = 45^\circ + 30^\circ, 105∘=60∘+45∘105^\circ = 60^\circ + 45^\circ.
  • Apply sum/difference formulas with known exact values.
  • Simplify radicals carefully (combine like terms under square roots).

Example: cos⁡15∘=cos⁡(45∘−30∘)=cos⁡45∘cos⁡30∘+sin⁡45∘sin⁡30∘=22⋅32+22⋅12=6+24\cos 15^\circ = \cos(45^\circ - 30^\circ) = \cos 45^\circ \cos 30^\circ + \sin 45^\circ \sin 30^\circ = \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2} = \frac{\sqrt{6} + \sqrt{2}}{4}

💡 Split weird angles into friendly pieces!

[EXEC: DEEP_COMPUTE]

3. Finding exact values for non-standard angles using addition/subtraction

Exact Values for Non-Standard Angles

Non-standard angles are those not among the common reference angles (0°, 30°, 45°, 60°, 90°). Exact trigonometric values for these angles are obtained by expressing them as sums or differences of standard angles, then applying sum/difference formulas.

This technique exploits the fact that any angle can be decomposed into combinations of angles with known exact values.

Core Rules:

  • Identify a decomposition using standard angles: e.g., 15∘=45∘−30∘15^\circ = 45^\circ - 30^\circ or 75∘=45∘+30∘75^\circ = 45^\circ + 30^\circ
  • Apply the appropriate sum/difference formula for the desired function
  • Simplify radicals to obtain the exact form
  • Multiple decompositions exist; choose the one yielding simplest computation

This method produces algebraically exact values without decimal approximation, crucial for theoretical work and precise calculations.

Example: sin⁡15∘=sin⁡(45∘−30∘)=sin⁡45∘cos⁡30∘−cos⁡45∘sin⁡30∘=22⋅32−22⋅12=6−24\sin 15^\circ = \sin(45^\circ - 30^\circ) = \sin 45^\circ \cos 30^\circ - \cos 45^\circ \sin 30^\circ = \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2} = \frac{\sqrt{6} - \sqrt{2}}{4}

TASK_1[0 / 3]
LVL_3
EXEC: FORMULA

Using the sum formula, the exact value of sin⁡(75∘)\sin(75^\circ) can be found by decomposing it as 75∘=45∘+30∘75^\circ = 45^\circ + 30^\circ. The final simplified exact value can be written in the form (a+b)/4(\sqrt a + \sqrt b)/4.

What is the value of the sum a+ba + b?

DEEP_COMPUTE
ULTRA
[EXEC: MICRO_CORE]

✖️ 4. Strategic proof and derivation techniques for complex trigonometric identities

🔍 Strategic Proof Techniques

  • Start with the more complex side and simplify toward the simpler side.
  • Use sum/difference formulas to expand compound angles.
  • Apply double-angle formulas to collapse 2θ2\theta terms.
  • Convert everything to sine and cosine if stuck with mixed functions.
  • Factor or combine fractions using common denominators.

Example: Prove sin⁡3θ=3sin⁡θ−4sin⁡3θ\sin 3\theta = 3\sin\theta - 4\sin^3\theta. Write sin⁡3θ=sin⁡(2θ+θ)=sin⁡2θcos⁡θ+cos⁡2θsin⁡θ\sin 3\theta = \sin(2\theta + \theta) = \sin 2\theta \cos\theta + \cos 2\theta \sin\theta, then substitute double-angle formulas and simplify.

💡 Expand, substitute, simplify—always work one side only!

[EXEC: DEEP_COMPUTE]

4. Strategic proof and derivation techniques for complex trigonometric identities

Proof Techniques for Trigonometric Identities

Complex trigonometric identities are proven by strategically applying sum, difference, and double-angle formulas to transform one side into the other. The process requires recognizing patterns and selecting appropriate substitutions.

Effective proof strategy involves working from the more complex side toward the simpler side, or transforming both sides to a common form.

Core Rules:

  • Express all functions in terms of sine and cosine when direct simplification fails
  • Apply sum/difference formulas to expand or collapse angle combinations
  • Use double-angle formulas to relate 2θ2\theta terms to θ\theta terms
  • Factor and combine fractions systematically; avoid expanding prematurely
  • Substitute Pythagorean identities (sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1) to eliminate squared terms

Mastery requires recognizing when to expand versus when to factor, and identifying which formula applies to the given structure.

Example: Prove sin⁡3θ=3sin⁡θ−4sin⁡3θ\sin 3\theta = 3\sin\theta - 4\sin^3\theta. Write sin⁡3θ=sin⁡(2θ+θ)=sin⁡2θcos⁡θ+cos⁡2θsin⁡θ=2sin⁡θcos⁡2θ+(1−2sin⁡2θ)sin⁡θ=3sin⁡θ−4sin⁡3θ\sin 3\theta = \sin(2\theta + \theta) = \sin 2\theta \cos\theta + \cos 2\theta \sin\theta = 2\sin\theta\cos^2\theta + (1-2\sin^2\theta)\sin\theta = 3\sin\theta - 4\sin^3\theta

TASK_1[0 / 3]
LVL_2
EXEC: FORMULA

Simplify the expression tan⁡x⋅cos⁡x\tan x \cdot \cos x to a single trigonometric function.

DEEP_COMPUTE
ULTRA
[EXEC: MICRO_CORE]

✖️ 5. Applications in projectile motion and wave interference

🚀 Applications in Physics

  • Projectile range: R=v2sin⁡2θgR = \frac{v^2 \sin 2\theta}{g} (maximum when θ=45∘\theta = 45^\circ since sin⁡90∘=1\sin 90^\circ = 1).
  • The sin⁡2θ\sin 2\theta term comes from combining horizontal and vertical motion equations.
  • Beat frequency: When two waves interfere, fbeat=∣f1−f2∣f_{beat} = |f_1 - f_2| creates amplitude modulation.
  • Sum formulas convert cos⁡(ω1t)+cos⁡(ω2t)\cos(\omega_1 t) + \cos(\omega_2 t) into product form showing beats.

Example: A ball launched at 20 meters per second at 30 degrees has range R=202sin⁡60∘10=400⋅0.86610≈34.6R = \frac{20^2 \sin 60^\circ}{10} = \frac{400 \cdot 0.866}{10} \approx 34.6 meters.

💡 Double-angle maximizes range; sum formulas reveal beats!

[EXEC: DEEP_COMPUTE]

5. Applications in projectile motion and wave interference

Physical Applications of Double-Angle Formulas

Double-angle formulas appear naturally in physics when analyzing systems with angular dependencies. In projectile motion, the range formula R=v2sin⁡2θgR = \frac{v^2 \sin 2\theta}{g} uses sin⁡2θ=2sin⁡θcos⁡θ\sin 2\theta = 2\sin\theta\cos\theta to relate launch angle to horizontal distance. In wave interference, beat frequency arises from the sum formula applied to waves of slightly different frequencies.

These applications demonstrate how trigonometric identities encode fundamental physical relationships.

Core Rules:

  • Projectile range is maximized when sin⁡2θ=1\sin 2\theta = 1, giving θ=45∘\theta = 45^\circ
  • Complementary angles (θ\theta and 90∘−θ90^\circ - \theta) yield equal ranges since sin⁡2θ=sin⁡(180∘−2θ)\sin 2\theta = \sin(180^\circ - 2\theta)
  • Beat frequency fbeat=∣f1−f2∣f_{beat} = |f_1 - f_2| emerges from cos⁡(A)−cos⁡(B)=−2sin⁡A+B2sin⁡A−B2\cos(A) - \cos(B) = -2\sin\frac{A+B}{2}\sin\frac{A-B}{2}
  • The double-angle structure reveals symmetries in physical systems

These formulas enable analytical solutions to optimization and interference problems.

Example: A projectile launched at 30 m/s and 30° has range R=(30)2sin⁡60∘10=900⋅3210≈77.9R = \frac{(30)^2 \sin 60^\circ}{10} = \frac{900 \cdot \frac{\sqrt{3}}{2}}{10} \approx 77.9 m

TASK_1[0 / 3]
LVL_2
EXEC: FORMULA

Two sound waves have frequencies of f1=440f_1 = 440 Hz and f2=444f_2 = 444 Hz. Calculate the beat frequency in Hz.

DEEP_COMPUTE
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