Solving Trigonometric Equations By Finding All Solutions

TL;DR

Learn how to find all solutions to trigonometric equations using reference angles and the unit circle.

Transcript

consider this equation sine x is equal to one-half what is the value of x so sine of what angle is equal to one-half if you use the unit circle you know that sine of 30 degrees or pi over six is equal to one-half and sine is positive in quadrants one and two so sine 5 pi over 6 is also equal to a half so therefore you could say x is equal to pi ove... Read More

Key Insights

• ❓ Sine is positive in quadrants one and two, while cosine is positive in quadrants one and four.
• 👨‍💼 Quadrants two and three yield negative values for both sine and cosine.
• ❎ Tangent is negative in quadrants two and four, and positive in quadrants one and three.
• ➕ The square root of a positive value is always positive, hence the plus or minus sign is required.
• 🖐️ Reference angles play a crucial role in finding solutions to trigonometric equations, especially when the function is positive or negative in specific quadrants.
• 🧡 The period of trigonometric functions determines the range of solutions and helps identify equivalent angles.
• 😑 Expressing solutions with the addition of 2πn or πn allows for writing a general equation that encompasses all possible solutions.

Questions & Answers

Q: How do you find the reference angle in trigonometric equations?

The reference angle is found by using the unit circle or the 30-60-90 triangle to determine the angle that corresponds to the given trigonometric value.

Q: What is the period of sine and cosine?

The period of sine and cosine functions is 2π, which means the function repeats every 2π radians or 360 degrees.

Q: How is the period of tangent and cotangent different from sine and cosine?

The period of tangent and cotangent functions is π, which means the function repeats every π radians or 180 degrees.

Q: How should solutions to trigonometric equations be expressed?

To represent all solutions, add 2πn or πn to the reference angle, where n is any integer. This accounts for the periodic nature of trigonometric functions.

Summary & Key Takeaways

• Trigonometric equations can be solved by finding the reference angle and adjusting it based on the quadrant where the function is positive or negative.

• The period of sine and cosine is 2π, while the period of tangent and cotangent is π.

• To find all solutions, add 2πn or πn to the reference angle, where n is any integer.