Problem 12 - Contemporary Precalculus

Question

In Exercises \(11-14\), assume that the terminal side of an angle of \(t\) radians passes through the given point on the unit circle. Find \(\sin t, \cos t,\) tan \(t\) $$(1 / \sqrt{10},-3 / \sqrt{10})$$

Step-by-Step Solution

Answer

Question: Given the point (1/√10, -3/√10) on the unit circle, find the values of sin t, cos t, and tan t. Answer: sin t = -3/√10, cos t = 1/√10, tan t = -3.

Step 1: Identify the point

The point on the unit circle is given as \((1/\sqrt{10}, -3/\sqrt{10})\).

Step 2: Find the sine value

The y-coordinate of the point on the unit circle corresponds to the sine value, so \(\sin t = -3/\sqrt{10}\).

Step 3: Find the cosine value

The x-coordinate of the point on the unit circle corresponds to the cosine value, so \(\cos t = 1/\sqrt{10}\).

Step 4: Find the tangent value

The tangent value can be found using the formula \(\tan t = \frac{\sin t}{\cos t}\). Substituting the previously found sine and cosine values, we get: \(\tan t = \frac{-3/\sqrt{10}}{1/\sqrt{10}} = -3\). So, the values of \(\sin t\), \(\cos t\), and \(\tan t\) are:

Step 5: Write down the results

- \(\sin t = -\dfrac{3}{\sqrt{10}}\) - \(\cos t = \dfrac{1}{\sqrt{10}}\) - \(\tan t = -3\)

Key Concepts

unit circle

The unit circle is a crucial tool in trigonometry, providing a simple and effective way to visualize trigonometric functions. A unit circle is a circle with a radius of one that is centered at the origin of the coordinate plane - Every point on this circle corresponds to an angle measured in radians - A point \(x, y\) on the unit circle helps to define the sine and cosine of an angle \(t\), where the x-coordinate of the point is \cos t\, and the y-coordinate is \sin t\. Because the radius of the unit circle is one, using the coordinates of a point on the circle simplifies the computation of trigonometric functions. The connection between the unit circle and these functions reflects the inherent geometry of circles and angles, making the unit circle an indispensable part of learning trigonometry.

sine and cosine

Sine and cosine are foundational trigonometric functions, often introduced through the unit circle. In the unit circle:
- The sine of an angle is represented by the y-coordinate of the corresponding point \( (x, y) \) on the circle - The cosine of an angle is the x-coordinate These coordinates represent side lengths of a right triangle formed by a line from the circle's origin to the point: - The angle \(t\) is the angle between this line and the positive x-axis It's beneficial to remember that sine and cosine values depend on the location of the angle on the unit circle - Angles in different quadrants may yield positive or negative values for sine and cosine- For example, in the second quadrant, sine values are positive, but cosine values are negative Understanding how sine and cosine translate into coordinates is key to mastering these functions.

tangent

Tangent is another core trigonometric function, derived from sine and cosine - The tangent of an angle \(t\) is defined as the ratio of the sine to the cosine: \tan t = \frac{\sin t}{\cos t}\. On the unit circle, tangent can be interpreted as the slope of the line that passes through the origin and the point representing that angle: - It illustrates how steeply this line inclines upwards or downwards Tangent has some unique properties:
- If \(\cos t = 0\), tangent is undefined because division by zero isn't possible - This happens when angles line up perfectly with the y-axis Tangent values can provide insights into the angle's behavior, especially in calculus and physics.