Problem 111 - Precalculus : Building Concepts and Connections

Question

This set of exercises will draw on the ideas presented in this section and your general math background. Find all values of \(t\) in \([0,2 \pi)\) such that \(\sin t=\cos t\)

Step-by-Step Solution

Answer

The values of \(t\) that satisfy \(\sin t = \cos t\) in the interval \([0,2\pi)\) are \(\frac{\pi}{4}\) and \(\frac{5\pi}{4}\).

Step 1: Use the trigonometric identities

We know from the trigonometric identities that for all \(t\), \(\sin(t) = \cos(\frac{\pi}{2} - t)\). Therefore, we should set up the following equation: \(\cos t = \sin(\frac{\pi}{2} - t)\).

Step 2: Find the equivalent terms

From the above equation, we can see that \(\sin t = \sin(\frac{\pi}{2} - t)\) is the equation we are looking to solve.

Step 3: Solve the equation

For two angles \(a\) and \(b\), if \(\sin a = \sin b\) then \(a = b + 2n\pi\) or \(a = \pi - b + 2n\pi\), where \(n\) is an integer. We use this property here: so \(\frac{\pi}{2}-t=t+2n\pi\) or \(\frac{\pi}{2}-t=\pi-t+2n\pi\)

Step 4: Find the values of \(t\)

From the first equation, we find \(t=\frac{\pi}{4}+n\pi\), from the second equation, we find \(t=\frac{\pi}{4}\) as \(n\) can be an integer, choose \(n=0,1\) to get \(t\) within \([0,2\pi]\), we find \(t=\frac{\pi}{4},\frac{5\pi}{4}\).

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the occurring variables where both sides of the equation are defined. These identities are useful in simplifying expressions and solving trigonometric equations. One key identity used in this exercise is that \(\sin(t) = \cos(\frac{\pi}{2} - t)\).

This identity is a manifestation of the cofunction identity which relates the sine and cosine functions. The cofunction identity states that the sine of an angle is equal to the cosine of its complement, where the complement of an angle \(\theta\) is \(\frac{\pi}{2} - \theta\). Leveraging this identity allows us to set up the equation discussed in the exercise, providing a clear pathway to find solutions for \(t\) which satisfy \(\sin t = \cos t\).

Sine and Cosine

The sine and cosine functions are fundamental in trigonometry, relating the angles of a right triangle to its side lengths. The sine function (sin) of an angle \(t\) in a right-angled triangle is the ratio of the length of the side opposite to \(t\) to the length of the hypotenuse. Conversely, the cosine function (cos) is the ratio of the adjacent side's length to the hypotenuse.

In the context of the unit circle, \(\sin(t)\) represents the y-coordinate, and \(\cos(t)\) represents the x-coordinate of a point on the unit circle at an angle \(t\) from the positive x-axis. Because these functions are periodic, they repeat their values in regular intervals, and this periodic nature is utilized to find multiple solutions in the given range.

Angle Measure in Radians

In mathematics, radians are the standard unit of angular measure. One radian is the angle created when the arc length of a circle is equal to the radius of the circle. This unit is especially prevalent in trigonometry as it simplifies many formulas.

The exercise provides the interval \(0, 2\pi)\) which refers to the angles from 0 to a full circle measured in radians. Since \(2\pi\) radians is equal to 360 degrees, finding values of \(t\) in this range is akin to exploring all possible angles in a circle, which is crucial in solving periodic trigonometric functions. The solution to the problem relies on understanding that radians provide a natural way to describe both the cyclical nature of trigonometric functions and the geometric properties that relate angles to lengths in circles.