91Ó°ÊÓ

The concept of inverse cosine (denoted as \(\text{cos}^{-1} x\)) is a fundamental topic in trigonometry. The inverse cosine function lets us find the angle whose cosine is a given number.
The notation \(\text{cos}^{-1} x\) might look like raising cosine to the power of -1, but that's not the case. It's actually asking: 'What angle, y, gives this value x when you take the cosine of y?'
For example, \(\text{cos}^{-1} \frac{1}{2} = \frac{\pi}{3}\), means that the cosine of \(\frac{\pi}{3}\) is \(\frac{1}{2}\). This concept is pivotal in solving trigonometric equations, especially when we need an angle as the answer.

Trigonometric identities are equations involving trigonometric functions that are true for every value of the variables involved. They play a crucial role in simplifying and solving trigonometric equations.
Here are some fundamental trigonometric identities:

• The Pythagorean Identity: \[ \text{sin}^2 \theta + \text{cos}^2 \theta = 1 \]
• The angle sum and difference identities:
• \(\text{cos} (a + b) = \text{cos} a \text{cos} b - \text{sin} a \text{sin} b\)
• \(\text{cos} (a - b) = \text{cos} a \text{cos} b + \text{sin} a \text{sin} b\)
• Double angle identities like:
• \(\text{cos} 2a = \text{cos}^2 a - \text{sin}^2 a\)

These identities help in transforming a trigonometric expression into a simpler one, making it easier to work with. For instance, in the given exercise, understanding that \(\text{cos}^{-1} x = y\) implies \(\text{cos} y = x\) is crucial to find the solution.

Solving trigonometric equations often involves using algebraic techniques and trigonometric identities. In our exercise, we follow these steps:
1. **Isolate the trigonometric function:** Here we divide both sides of the equation by 2 to isolate the inverse cosine function.
2. **Apply the definition of the inverse trigonometric function:** We know that \(\text{cos}^{-1} x = y\) means \(\text{cos} y = x\). Using this information helps us set up the equation correctly.
3. **Find the exact value:** Utilizing known trigonometric values, such as \(\text{cos} \frac{\text{\pi}}{2} = 0\), allows us to solve for x easily.

By following these steps systematically, we can solve more complex trigonometric equations involving different functions and identities. Understanding these processes is key to mastering trigonometry.