TOPICS: Trigonometry - Sum and Difference of Arcs

Keywords

• Sum of arcs
• Difference of arcs
• Addition formula
• Subtraction formula
• Sine (sin)
• Cosine (cos)
• Tangent (tan)
• Notable angles
• Trigonometric identities

Key Questions

• How to apply the formula of sum of arcs for sine, cosine, and tangent?
• How to apply the formula of difference of arcs for sine, cosine, and tangent?
• How to calculate the sine, cosine, and tangent of non-standard angles, such as 75º and 22.5º?
• In what practical situations do we use the sum and difference of arcs?

Crucial Topics

• Understanding and applying the formulas of sum and difference of arcs.
• Relationship between complementary and supplementary arcs.
• Simplification of trigonometric expressions using identities.

Formulas

Sum of Arcs Formula

• Sine: $ \sin(\alpha + \beta) = \sin(\alpha)\cos(\beta) + \cos(\alpha)\sin(\beta) $
• Cosine: $ \cos(\alpha + \beta) = \cos(\alpha)\cos(\beta) - \sin(\alpha)\sin(\beta) $
• Tangent: $ \tan(\alpha + \beta) = \frac{\tan(\alpha) + \tan(\beta)}{1 - \tan(\alpha)\tan(\beta)} $

Difference of Arcs Formula

• Sine: $ \sin(\alpha - \beta) = \sin(\alpha)\cos(\beta) - \cos(\alpha)\sin(\beta) $
• Cosine: $ \cos(\alpha - \beta) = \cos(\alpha)\cos(\beta) + \sin(\alpha)\sin(\beta) $
• Tangent: $ \tan(\alpha - \beta) = \frac{\tan(\alpha) - \tan(\beta)}{1 + \tan(\alpha)\tan(\beta)} $

Remember: Master of formulas, master of solutions!

NOTES: Theme Detailing

Key Terms

• Sine (sin): Relationship between the opposite side and the hypotenuse in a right triangle.
• Cosine (cos): Relationship between the adjacent side and the hypotenuse in a right triangle.
• Tangent (tan): Ratio between sine and cosine, or the opposite side over the adjacent side.
• Notable angles: Angles with known or easily derivable sine, cosine, and tangent values, such as 30º, 45º, and 60º.
• Trigonometric identities: Equations that remain true for all values inserted into them, such as the sum and difference of angles formulas.

Main Ideas

• Understanding sum and difference of arcs is crucial to simplify trigonometric expressions and calculate values of non-standard angles.
• The sum and difference formulas are powerful tools to solve problems involving composition of movements or waves.

Topic Contents

• Sum of Arcs Formula:
  • Explains how the sum of trigonometric functions can be decomposed.
  • Allows for easy calculation of an unknown angle from the sum of known angles.
• Difference of Arcs Formula:
  • Analogous to the sum formula, but applied to the difference between angles.
  • Used to find the trigonometric function of an angle by subtracting two known angles.

Examples and Cases

• Calculating the Sine of 75º:
  • We know that 75º = 45º + 30º.
  • We can use the sum formula: $ \sin(75º) = \sin(45º)\cos(30º) + \cos(45º)\sin(30º) $.
  • We substitute the known values: $ \sin(75º) = \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2} $.
  • We simplify to reach the final result.
• Calculating the Cosine of 22.5º:
  • We use the difference of arcs recognizing that 22.5º = 45º - 22.5º.
  • We apply the difference formula: $ \cos(22.5º) = \cos(45º)\cos(22.5º) + \sin(45º)\sin(22.5º) $.
  • The challenge is to calculate $\cos(22.5º)$ and $\sin(22.5º)$, which in turn can be found using the angle bisector formula (an advanced topic not covered here).

Remember: Practice makes perfect - practice the formulas with different examples until the process becomes intuitive!

SUMMARY: Consolidating Trigonometry of Sum and Difference of Arcs

Summary of the most relevant points

• Effective methods: Using the sum and difference of arcs formulas to find sine, cosine, and tangent of non-standard angles is an effective strategy in trigonometry.
• Crucial identities: The sum and difference identities are essential to simplify and solve complex trigonometric expressions.
• Notable angles and their relations: Decomposing unknown angles into sums or differences of notable angles (30º, 45º, 60º) facilitates the calculation of their trigonometric functions.
• Addition and subtraction formulas: Understanding the mechanics behind the addition formulas ($ \sin(\alpha + \beta) $, $ \cos(\alpha + \beta) $, $ \tan(\alpha + \beta) $) and subtraction ($ \sin(\alpha - \beta) $, $ \cos(\alpha - \beta) $, $ \tan(\alpha - \beta) $) is fundamental.

Conclusions

• The sum and difference formulas can be applied to calculate the trigonometric functions of complex angles, such as 75º and 22.5º, using the sum or subtraction of simpler and known angles.
• Constant practice in applying these formulas leads to mastery of trigonometric concepts and enhances the ability to solve a variety of practical problems.
• Understanding patterns and relationships in trigonometry, such as identities and angle relations, is crucial for efficiently solving problems.
• Learning trigonometry of sum and difference of arcs is a progressive skill, developed through studying examples, solving exercises, and exploring practical cases.

Remember: Arming yourself with the correct formulas is half the battle!