Precalculus challenge: can we just cancel out the sine?

TL;DR

This video explains how to solve equations involving sine and cosine identities using the double angle identity and the reference triangle method.

Transcript

okay this is the challenge for all the precalculus students we have two equations on the spot the first one is sine of two theta times equal to sine theta and for the second one we have sine of pi theta that's equal to sine theta maybe for the first point you are thinking to use the double angle density for sine two theta sure but can you do the si... Read More

Key Insights

• 👨‍💼 The double angle identity for sine is 2sin(θ)cos(θ).
• 👨‍💼 Equations involving sine and cosine identities may have infinitely many solutions.
• 🔺 Reference triangles can be used to find solutions to equations involving cosine and πθ.
• 👨‍💼 Canceling out the sine is not valid in equations with different inputs.
• 🔬 Brilliant is a recommended resource for learning and practicing math and science concepts.

Questions & Answers

Q: How do you use the double angle identity to solve equations with sine?

To solve an equation like sine(2θ) = sine(θ), apply the double angle identity, which states that sine(2θ) = 2sin(θ)cos(θ). Rearrange the equation and consider different values of θ to find all possible solutions.

Q: How do you solve equations involving cosine and πθ?

Equations like cosine(πθ) = cosine(θ) require considering cases and adding odd multiples of π to the angles. By doing so, you can find all the solutions to the equation.

Q: Can you cancel out the sine in equations like sine(πθ) = sine(θ)?

No, you cannot cancel out the sine in equations with different inputs. The video explains that canceling out the sine is not valid because sine is not injective. Instead, you need to consider different cases and add multiples of π to the angles.

Q: Where can I learn more about solving equations with trigonometric identities?

The video suggests checking out Brilliant, a problem-solving website and app that offers courses in math, science, and computer science. They have courses that can help you deepen your understanding of trigonometric identities and other topics.

Summary & Key Takeaways

• The video introduces two equations involving sine and cosine: sine(2θ) = sine(θ) and cosine(πθ) = cosine(θ).

• The traditional way to solve the first equation is to use the double angle identity, which results in infinitely many solutions.

• The second equation requires considering different cases and adding odd multiples of π to the angles.