91Ó°ÊÓ

To solve the given trigonometric equation, it's crucial to understand the sine function, \( \sin(\theta) \). The sine function relates an angle in a right triangle to the ratio of the length of the opposite side over the hypotenuse. This can be expressed as:
\( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \).

Sine is periodic, repeating every \(2\pi\) radians (360 degrees). This means that \( \sin(\theta + 2\pi) = \sin(\theta) \). Keep this periodic nature of sine in mind when solving any trigonometric equation.

In the unit circle, the sine of an angle corresponds to the y-coordinate of the point where the terminal side of the angle intersects the circle.
Consider an angle \( \theta \) in the first quadrant; the sine of \( \theta \) is positive, and as \( \theta \) moves to the second quadrant, sine still remains positive. Conversely, in the third and fourth quadrants, sine is negative. This helps determine the values of \( \theta \) when solving \( \sin(\theta) \) equations within a specific interval.

Next, let's explore the cosine function, \( \cos(\theta) \). Much like the sine function, the cosine relates an angle to the ratio of the length of the adjacent side over the hypotenuse in a right triangle:
\( \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \).

Cosine is also a periodic function, with a period of \(2\pi\). This means \( \cos(\theta + 2\pi) = \cos(\theta) \).

On the unit circle, the cosine of an angle \( \theta \) is represented by the x-coordinate of the point where the terminal side of the angle intersects the circle.

In the first and fourth quadrants, cosine values are positive. In contrast, in the second and third quadrants, cosine values are negative. This information helps narrow down potential solutions when solving equations involving \( \cos(\theta) \).

Trigonometric identities form the backbone of solving trigonometric equations. They allow us to simplify expressions and find solutions more easily. Here are some key identities that are essential to know:

• Pythagorean Identity: \( \sin^2(\theta) + \cos^2(\theta) = 1 \)


• Angle Sum and Difference Identities:
  \( \sin(a \pm b) = \sin(a)\cos(b) \pm \cos(a)\sin(b) \)
  \( \cos(a \pm b) = \cos(a)\cos(b) \mp \sin(a)\sin(b) \)


• Double Angle Identities:
  \( \sin(2a) = 2\sin(a)\cos(a) \)
  \( \cos(2a) = \cos^2(a) - \sin^2(a) \)


• Half Angle Identities:
  \( \sin^2\left(\frac{a}{2}\right) = \frac{1 - \cos(a)}{2} \)
  \( \cos^2\left(\frac{a}{2}\right) = \frac{1 + \cos(a)}{2} \)

In our given exercise, since we have \(\sin(\theta) + \cos(\theta) = \sqrt{2} \), we use the identity:
\((\sin(\theta) + \cos(\theta))^2 = \sin^2(\theta) + \cos^2(\theta) + 2\sin(\theta) \cos(\theta) \).

By recognizing that \( \sin^2(\theta) + \cos^2(\theta) = 1 \) and setting up the equation accordingly, we can isolate and solve for \( \theta \) accurately.