Half Angle Formulas & Identities - Evaluating Trigonometric Expressions | Summary and Q&A

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October 19, 2017
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The Organic Chemistry Tutor
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Half Angle Formulas & Identities - Evaluating Trigonometric Expressions

TL;DR

The content explains the derivation and usage of the half angle formulas for sine, cosine, and tangent.

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Key Insights

  • 🔺 The half angle formulas for sine, cosine, and tangent are derived from the power reducing formulas and involve dividing the angles by 2 and taking the square root.
  • 🔺 The half angle formulas allow us to calculate the exact values of trigonometric functions for certain angles.
  • 🔺 The formulas can be used to find the half angle of any given angle, and the positive or negative sign should be chosen based on the quadrant of the given angle.

Transcript

now starting with the power reducing formula of sine squared we're going to get the half angle formula so sine squared we know it's 1 minus cosine 2 theta divided by two so that's true then sine squared theta divided by two must be equal to one minus cosine theta over two so if we divide this angle by 2 then we should divide that angle by 2. now al... Read More

Questions & Answers

Q: What is the half angle formula for sine?

The half angle formula for sine is sine (theta/2) = ±√(1 - cos(theta))/2.

Q: How can the half angle formulas for sine and cosine be derived?

The derivation involves starting with the power reducing formulas for sine squared and cosine squared, dividing the angles by 2, and taking the square root of both sides.

Q: What is the half angle formula for tangent?

The half angle formula for tangent can be calculated using the formulas for sine and cosine. It is tangent(theta/2) = ±√((1 - cos(theta))/(1 + cos(theta))) = (1 - cos(theta))/sin(theta) = sin(theta)/(1 + cos(theta)).

Q: How do you evaluate cosine(15 degrees) using the half angle formula?

By setting theta/2 as 15 degrees and solving for theta = 30 degrees. Plugging in the value, cosine(15 degrees) = ±√(1 + cos(30 degrees))/2 = ±(√2 + √3)/2.

Q: What is the exact value of sine(22.5 degrees) using the half angle formula?

By setting theta/2 as 22.5 degrees and solving for theta = 45 degrees. Plugging in the value, sine(22.5 degrees) = √(1 - cos(45 degrees))/2 = (√2 - √2)/2.

Q: How can tangent(75 degrees) be evaluated using the half angle formula?

By setting theta/2 as 75 degrees and solving for theta = 150 degrees. Plugging in the value, tangent(75 degrees) = (1 - cos(150 degrees))/sin(150 degrees) = 2 + √3.

Summary & Key Takeaways

  • The content discusses the derivation of the half angle formula for sine and cosine.

  • It explains how to calculate the half angle of tangent using the derived formulas.

  • The content also provides examples of evaluating specific trigonometric functions using the half angle formulas.

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