Question

If \(\theta\) is an acute angle, solve the equation \(\cos \theta=\frac{\sqrt{2}}{2}\).

Short answer

\(\theta = 45^\circ\) or \(\theta = \frac{\pi}{4}\).

Step-by-step solution

Recall Fundamental Properties

Recall the trigonometric values for common angles. The value \(\frac{\sqrt{2}}{2}\) is a common trigonometric value that corresponds to specific angles.

Identify the Reference Angle

Determine which angle \(\theta\) has a cosine of \(\frac{\sqrt{2}}{2}\). In the unit circle, \(\theta = 45^\circ\) (or \(\theta = \frac{\pi}{4}\)) satisfies this condition.

Verify the Acute Angle Condition

Since \(\theta\) is given to be an acute angle, confirm that the identified angle aligns with the acute angle definition, i.e., \(\theta < 90^\circ\) or \(\theta < \frac{\pi}{2}\). For \(\theta = 45^\circ\), this condition is true.

Write the Solution

Conclude that the angle which satisfies the equation \(\cos \theta = \frac{\sqrt{2}}{2}\) and is an acute angle is \(\theta = 45^\circ\) or \(\theta = \frac{\pi}{4}\).

Key concepts

Cosine Function

In trigonometry, the cosine function is one of the primary trigonometric functions. It is abbreviated as \(\cos\). The cosine of an angle in a right triangle is defined as the ratio of the length of the adjacent side to the length of the hypotenuse. If we have a right triangle with an angle \(\theta\), the cosine function can be written as: \[\cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}}\] The cosine function is particularly useful in various branches of mathematics and engineering. It helps in solving equations and finding angles in geometric shapes. On the unit circle, the cosine of an angle \(\theta\) is the \(x\)-coordinate of the point where the terminal side of the angle intersects the circle.

Acute Angles

An acute angle is any angle that is less than 90 degrees or \(\frac{\pi}{2}\) radians. These angles are smaller than right angles and are often encountered in trigonometric problems. In our specific problem, we were asked to find an acute angle \(\theta\) such that \(\cos \theta = \frac{\sqrt{2}}{2}\). Since \(45^\circ\) (or \(\frac{\pi}{4}\)) is less than \(90^\circ\), it qualifies as an acute angle. Other common acute angles include 30 degrees (or \(\frac{\pi}{6}\)) and 60 degrees (or \(\frac{\pi}{3}\)). These angles frequently appear in problems involving the unit circle and basic trigonometry.

Unit Circle

The unit circle is a fundamental concept in trigonometry and serves as a powerful tool for understanding trigonometric functions. It is a circle with a radius of one unit centered at the origin of the coordinate plane. The coordinates of any point on the unit circle can be represented as \( (\cos \theta, \sin \theta) \), where \(\theta\) is the angle formed with the positive \(x\)-axis. The unit circle allows us to find the sine and cosine of specific angles easily. For example, when the angle \(\theta\) is \(45^\circ\) (or \(\frac{\pi}{4}\)), the point at which the terminal side of the angle intersects the unit circle is \(\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)\). Therefore, \(\cos 45^\circ = \frac{\sqrt{2}}{2}\) and \(\sin 45^\circ = \frac{\sqrt{2}}{2}\). Using the unit circle, we can solve a variety of trigonometric equations by visualizing and understanding the position of angles and their corresponding coordinates.