Problem 30 - Contemporary Precalculus

Question

Assume that \(\sin t=3 / 5\) and \(0< t <\pi / 2 .\) Use identities in the text to find the number. $$\sin (t+10 \pi)$$

Step-by-Step Solution

Answer

Question: Given that \(\sin t = \frac{3}{5}\) and \(0 < t < \frac{\pi}{2}\), find the value of \(\sin(t + 10\pi)\). Answer: \(\sin(t + 10\pi) = \frac{3}{5}\)

Step 1: Find the cosine of t

Since we know \(\sin t = \frac{3}{5}\) and \(t\) is in the first quadrant (\(0 < t < \frac{\pi}{2}\)), we can find the cosine value using the Pythagorean identity, \(\sin^2 t + \cos^2 t = 1\): $$ \cos^2 t = 1 - \sin^2 t = 1 - \left(\frac{3}{5}\right)^2 $$ Now solve for \(\cos t\): $$ \cos t = \sqrt{1 - \left(\frac{3}{5}\right)^2} = \frac{4}{5} $$ Since \(t\) is in the first quadrant, the cosine is positive, so \(\cos t = \frac{4}{5}\).

Step 2: Find the cosine of \(10\pi\)

To find the cosine of \(10\pi\), we need to find an angle coterminal to \(10\pi\) (an angle that terminates at the same point on the unit circle). We know that the cosine function has a period of \(2\pi\), so adding or subtracting multiples of \(2\pi\) will result in the same cosine value. \(10\pi = 5(2\pi)\), which means the angle \(10\pi\) is coterminal with \(0\). Since \(\cos 0 = 1\), we have \(\cos 10\pi = 1\).

Step 3: Use the Angle Sum Identity

We now have all the values needed to use the angle sum identity \(\sin(A+B) = \sin A\cos B + \cos A\sin B\). Substitute the values into the identity: $$ \sin(t + 10\pi) = \sin t \cos 10\pi + \cos t \sin 10\pi $$ We know \(\sin t = \frac{3}{5}\), \(\cos t = \frac{4}{5}\), and \(\cos 10\pi = 1\). To find \(\sin 10\pi\), recall that its coterminal angle is \(0\), and \(\sin 0 = 0\). Plug in the values: $$ \sin(t + 10\pi) = \left(\frac{3}{5}\right)(1) + \left(\frac{4}{5}\right)(0) $$

Step 4: Simplify

Simplify the expression to find the final value of \(\sin(t + 10\pi)\): $$ \sin(t + 10\pi) = \frac{3}{5} $$ So, \(\sin(t + 10\pi) = \frac{3}{5}\).

Key Concepts

sin and cosine

Sine (\( \sin \) ) and cosine (\( \cos \) ) are fundamental trigonometric functions that describe the relationship between the angles and lengths in right-angled triangles. Each function corresponds to a specific ratio formed in a right-angled triangle:

• Sine: The sine of an angle is the ratio of the length of the opposite side to the hypotenuse. In formula terms, this is \( \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} \)
• Cosine: The cosine of an angle is the ratio of the length of the adjacent side to the hypotenuse. This is expressed as \( \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} \)

These ratios are crucial for determining the size of an angle in a triangle, often requiring the Pythagorean identity to bridge them. The Pythagorean identity is given as \( \sin^2 \theta + \cos^2 \theta = 1 \).
In the given exercise, you identified \( \sin t = \frac{3}{5} \).Using the Pythagorean identity helps you find \( \cos t \) by:

• Calculating \( \cos^2 t = 1 - \left( \frac{3}{5} \right)^2 \)
• Then determining \( \cos t = \frac{4}{5} \)

Since the angle \( t \) was in the first quadrant, both sine and cosine are positive.

angle sum identity

Trigonometric identities like the angle sum identity are invaluable tools in mathematics. They allow us to express trigonometric functions of sums and differences of angles, helping solve complex equations. The angle sum identity for sine is:\[ \sin(A + B) = \sin A \cos B + \cos A \sin B \]This equation lets us calculate the sine of the sum of two angles, \( A \) and \( B \),by knowing the sine and cosine of each individual angle.
In our problem, this is applied to \( \sin(t + 10\pi) \). Given:

• \( \sin t = \frac{3}{5} \)
• \( \cos 10\pi = 1 \)
• \( \sin 10\pi = 0 \)
• \( \cos t = \frac{4}{5} \)

By substituting these into the identity
we compute:\[ \sin(t + 10\pi) = \left( \frac{3}{5} \right)(1) + \left( \frac{4}{5} \right)(0) \]This shows how the identity simplifies calculations involving angle sums.

unit circle

The unit circle is an essential concept in trigonometry. It's a circle with a radius of 1 unit. It's centered at the origin of a coordinate plane. This circle helps visualize trigonometric functions and their relationships.

• The x-coordinate of a point on the unit circle represents \( \cos \theta \)
• The y-coordinate stands for \( \sin \theta \)

An angle \( \theta \) in standard position has its vertex at the origin, and its initial side along the positive x-axis with its terminal side completing the angle inside the circle.
A key characteristic of the unit circle is how trigonometric functions repeat their values in regular intervals. For example, adding \( 2\pi \) to any angle gives an angle coterminal with the original. This means the cosine and sine values remain the same, as seen withthe angle \( 10\pi \), which is coterminal with \( 0 \). Therefore, \( \cos 10\pi = 1 \) and \( \sin 10\pi = 0 \), showing these values remain consistent each full circle rotation. Such properties simplify many trigonometric problems.