91Ó°ÊÓ

The double-angle identity is a crucial tool in trigonometry. It allows you to express trigonometric functions of double angles in terms of single angles.
In this exercise, we used the double-angle identity for cosine, which states: \(\text{cos}(2 \theta) = 1 - 2 \text{sin}^2(\theta)\).
This identity helps to transform the original equation \(\text{cos}(2 \theta) + 6 \text{sin}^2(\theta) = 4\) into an equation involving only sine.
These identities are very useful for simplifying and solving equations in trigonometry.

When encountering trigonometric equations, always look for opportunities to use identities like the double-angle identity. They can significantly simplify complex problems.

Trigonometric identities are like the 'magic keys' for solving trig equations. They allow you to rewrite expressions in different forms, making it easier to solve equations.
Common trigonometric identities include:

• Pythagorean identities: \(\text{sin}^2(\theta) + \text{cos}^2(\theta) = 1\).
• Angle sum and difference identities like: \(\text{sin}(a \pm b) = \text{sin}(a)\text{cos}(b) \pm \text{cos}(a)\text{sin}(b)\).
• Double-angle identities such as: \(\text{cos}(2 \theta) = 2\text{cos}^2(\theta) - 1\).

In this exercise, we specifically used the double-angle identity for cosine. Once you have the equation in a simpler form, you can solve for the trigonometric function's values.
Learn these identities by heart, and with practice, you’ll find them second nature when solving problems.

When solving trigonometric equations, it's crucial to consider the interval in which you need to find solutions. In this case, the interval is \(0 \leq \theta < 2 \pi\). This specifies the range of angles we are interested in.

After solving for the trigonometric function's values, we need to identify all possible angles within the given interval.
For \(\text{sin}(\theta) = \frac{\text{sqrt}(3)}{2}\), the values of \(\theta\) are \( \frac{\text{pi}}{3} \) and \( \frac{2\text{pi}}{3} \).
For \(\text{sin}(\theta) = -\frac{\text{sqrt}(3)}{2}\), the values of \(\theta\) are \(\frac{4\text{pi}}{3}\) and \( \frac{5\text{pi}}{3} \).

Understanding and applying interval solutions help ensure that you don’t miss any possible values that satisfy the equation within the specified range.