For which angles ¦È,
with 0¡ã ¡Ü ¦È ¡Ü 180¡ã, is sin 3¦È = -1?
How do I set this up?
To set up the problem, you need to solve the equation sin 3¦È = -1 for the values of ¦È within the given range of 0° ≤ ¦È ≤ 180°.
Start by rearranging the equation: sin 3¦È = -1 can be rewritten as sin 3¦È + 1 = 0.
Next, use the property of the sine function: sin (180° - x) = sin x. This means that if ¦È is a solution, then (180° - ¦È) is also a solution.
Now, we have the equation sin 3¦È + 1 = 0. You can solve this equation by factoring or using trigonometric identities. However, since this equation does not factor easily, we will use a trigonometric identity.
The triple-angle identity for sine states that sin 3x = 3sin x - 4(sin x)^3. Applying this identity to the equation, we get:
3sin ¦È - 4(sin ¦È)^3 + 1 = 0.
Now, we have a cubic equation. To solve this equation, you can use various methods such as factoring, using the Rational Root Theorem, or using a graphing calculator.
Once you find the values of ¦È that satisfy the equation in the range of 0° ≤ ¦È ≤ 180°, remember to also include their corresponding values (180° - ¦È) since they are also solutions.
To summarize:
1. Rewrite the equation as sin 3¦È + 1 = 0.
2. Use the triple-angle identity sin 3x = 3sin x - 4(sin x)^3 to simplify the equation.
3. Solve the resulting cubic equation using appropriate methods.
4. Identify the values of ¦È within the given range that satisfy the equation.
5. Include the corresponding values (180° - ¦È) as solutions.