Problem 32 - Precalculus Essentials

Question

\(0 \leq t<\frac{\pi}{2}\) and \(\sin t\) is given. Use the Pythagorean identity \(\sin ^{2} t+\cos ^{2} t=1\) to find \(\cos t\) $$\sin t=\frac{\sqrt{21}}{5}$$

Step-by-Step Solution

Answer

The result is \(\cos t = \frac{2}{5}\)

Step 1: Substitute the given sin value

Substitute \(\sin t = \frac{\sqrt{21}}{5}\) into Pythagorean identity \(\sin^2 t + \cos^2 t = 1\). The result is \((\frac{\sqrt{21}}{5})^2 + \cos^2 t = 1\).

Step 2: Solving for cos(t)

Next, rearrange the equation in order to solve for \(\cos t\) which results in \(\cos^2 t = 1 - (\frac{\sqrt{21}}{5})^2\)

Step 3: Simplify the equation and take the square root

After performing the calculation, \(\cos^2 t = 1 - \frac{21}{25}\) becomes \(\cos^2 t = \frac{4}{25}\). Taking the square root on both sides gives \(\cos t = \pm \frac{2}{5}\). But because \(0 \leq t < \frac{\pi}{2}\), \(\cos t\) must be positive. Hence, \(\cos t = \frac{2}{5}\).

Key Concepts

Pythagorean identity

One of the most fundamental relationships in trigonometry is the Pythagorean identity. This identity is expressed as \[ \sin^2 t + \cos^2 t = 1\] It tells us that for any angle \( t \), the square of the sine of \( t \) plus the square of the cosine of \( t \) will always equal one. This identity is so central because it directly relates the sine and cosine of an angle through the concept of a unit circle.
It's important to understand how this identity comes into play with a right triangle. In a right triangle, the hypotenuse is 1 in the unit circle, sin corresponds to the opposite side, and cosine corresponds to the adjacent side.
Using the Pythagorean identity allows us to find one trigonometric ratio when we know the other. In the given exercise, knowing \( \sin t \) helps us find \( \cos t \) using this identity directly.
Whenever calculating other trigonometric values like tangent and their inverses, the Pythagorean identity acts as a bridge to simplify and solve problems.

sine and cosine relationship

The sine (sin) and cosine (cos) functions are perhaps the most important in trigonometry. Understanding their relationship is key to mastering trigonometric concepts. Given an angle \( t \) in a right triangle or a unit circle, sine and cosine are defined as:

• \( \sin t \) is the ratio of the length of the side opposite the angle \( t \) to the hypotenuse.
• \( \cos t \) is the ratio of the length of the adjacent side to the hypotenuse.

These two functions are tightly related through the Pythagorean identity. They ensure that even when angles change, this relationship stays constant and balanced.
In different quadrants of the unit circle, the signs of \( \sin t \) and \( \cos t \) can change, but the identity's balance remains unaffected. In our exercise, since \( 0 \leq t < \frac{\pi}{2} \), both \( \sin t \) and \( \cos t \) are positive, and this aids in deciding the correct sign for \( \cos t \) after solving with the identity.Understanding sine and cosine not just as separate functions but as parts of a cohesive system is essential for all trigonometric applications.

trigonometric identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the variables where both sides of the equation are defined. Besides the Pythagorean identity, there are many other important identities in trigonometry such as angle sum and difference identities, double angle formulas, and reciprocal identities like:

• \( \tan t = \frac{\sin t}{\cos t} \)
• \( \sec t = \frac{1}{\cos t} \)
• \( \csc t = \frac{1}{\sin t} \)

These identities are crucial for simplifying expressions and solving trigonometric equations. They bridge the gaps between different trigonometric functions and allow for transformations across various forms.
The beauty of trigonometric identities lies in their ability to reframe problems in accessible formats. Often, complex trigonometric problems become manageable simply by applying the right identity.
In practical situations, these identities are used to create models and solve real-world scenarios involving waves, oscillations, and circles among others. It's the interconnection and application of these identities that make trigonometry a powerful mathematical tool.