Right Triangle Trigonometry and Half Angle Identities & Formulas

TL;DR

Find the values of sine theta/2 and cosine theta/2 using a right triangle and trigonometric formulas.

Transcript

now let's say if you're given a right triangle and here you have the angle theta and this is 3 4 and 5. using this information find the value of sine theta divided by two so let's write the formula first this is going to be plus or minus the square root of one minus cosine theta divided by two now x and y are both positive therefore this triangle i... Read More

Key Insights

• 🗯️ Trigonometric formulas and a right triangle can be used to find the values of sine theta/2, cosine theta/2, and tangent theta/2.
• 😌 The values obtained for sine theta/2, cosine theta/2, and tangent theta/2 depend on the characteristics of the given angle and the quadrant it lies in.
• ❓ Multiple formulas can be employed to find the value of tangent theta/2, providing different approaches for calculation and verification.

Questions & Answers

Q: How can the value of sine theta/2 be found using a right triangle?

By using the formula sine theta/2 = sqrt(1 - (cosine theta/2)^2), where cosine theta/2 can be found using the adjacent side and hypotenuse of the right triangle.

Q: What if the given angle is between pi/2 and pi? How does that affect the values of sine theta/2 and cosine theta/2?

In this case, the values of sine theta/2, cosine theta/2, and tangent theta/2 will all be positive, as the angle is in quadrant one. The formulas and calculations remain the same.

Q: Can another formula be used to find tangent theta/2?

Yes, another formula for tangent theta/2 is derived, which involves the values of sine theta/2 and cosine theta/2. Both approaches yield the same result.

Q: How can the values of sine theta/2 and cosine theta/2 be confirmed using the given tangent value?

By using the equation sine/(1 + cosine), with sine theta being the given value and cosine theta computed from the triangle, the resulting calculation should yield the given tangent value if correct.

Summary & Key Takeaways

• The video explains how to find the value of sine theta/2 and cosine theta/2 using a right triangle with a given angle and side lengths.

• By using the formulas for sine and cosine in a right triangle, the video demonstrates how to calculate these values.

• In a second example, the video shows how to find the values of sine theta/2, cosine theta/2, and tangent theta/2 when only given the tangent value and the range of the angle.