Problem 41 - Contemporary Precalculus

Question

In Exercises \(37-42,\) find \(\sin t,\) cos \(t,\) tan \(t\) when the terminal side of an angle of t radians in standard position passes through the given point. $$(\sqrt{3},-8)$$

Step-by-Step Solution

Answer

Question: Given that the terminal side of angle 𝑡 in standard position passes through the point \((\sqrt{3},-8)\), find sin(𝑡), cos(𝑡), and tan(𝑡). Answer: To find the sine, cosine, and tangent of angle 𝑡, we calculated the values of r, x, and y, using the Pythagorean theorem, and found that $$\sin t = \frac{-8}{\sqrt{67}}$$ $$\cos t = \frac{\sqrt{3}}{\sqrt{67}}$$ $$\tan t = \frac{-8}{\sqrt{3}}$$

Step 1: Calculate the distance r to the given point

To calculate the distance r from the origin to the point \((\sqrt{3},-8)\), use the Pythagorean theorem: $$r = \sqrt{x^2 + y^2}$$ In our case, \(x = \sqrt{3}\) and \(y = -8\). Therefore, $$r = \sqrt{(\sqrt{3})^2 + (-8)^2} = \sqrt{3 + 64} = \sqrt{67}$$

Step 2: Determine the values of x and y

The x-coordinate of the given point is \(\sqrt{3}\), so \(x = \sqrt{3}\). The y-coordinate is -8, so \(y = -8\).

Step 3: Calculate sin(t), cos(t), and tan(t)

Now that we have the values of r, x, and y, we can use the definitions of sine, cosine, and tangent to calculate sin(𝑡), cos(𝑡), and tan(𝑡): $$\sin t = \frac{y}{r} = \frac{-8}{\sqrt{67}}$$ $$\cos t = \frac{x}{r} = \frac{\sqrt{3}}{\sqrt{67}}$$ $$\tan t = \frac{y}{x} = \frac{-8}{\sqrt{3}}$$ So, \(\sin t = \frac{-8}{\sqrt{67}}\), \(\cos t = \frac{\sqrt{3}}{\sqrt{67}}\), and \(\tan t = \frac{-8}{\sqrt{3}}\).

Key Concepts

Understanding the Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry that states in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

This theorem can be written as: \( c^2 = a^2 + b^2 \) where \( c \) is the length of the hypotenuse, and \( a \) and \( b \) are the lengths of the triangle's other two sides. In the context of trigonometry and the exercise at hand, we use the Pythagorean theorem to find the distance from the origin to a point on a Cartesian plane, which is the hypotenuse of a right-angled triangle formed by the point's coordinates.

Using this theorem as a stepping stone, we can explore and understand the trigonometric functions of sine, cosine, and tangent, which relate the angles of a triangle to the lengths of its sides.

Decoding the Sine Function

The sine function, often abbreviated as \( \sin \), is a trigonometric function that relates a given angle to the ratio of the length of the side opposite that angle to the length of the hypotenuse in a right-angled triangle.

Mathematically, it is expressed as: \( \sin(\theta) = \frac{\text{opposite side}}{\text{hypotenuse}} \). For an angle \( t \) whose terminal side passes through \( (\sqrt{3}, -8) \) on the Cartesian plane, we first determine the distance from the origin to the point. This distance serves as the hypotenuse of the right-angled triangle, and the y-coordinate (-8) represents the length of the side opposite to angle \( t \).

To calculate \( \sin(t) \) for the given point, you divide the y-coordinate by the length of the hypotenuse. The result is a negative value because the point lies below the x-axis, indicating an angle measuring more than 180 degrees but less than 270 degrees, or in the third quadrant of the unit circle.

Interpreting the Cosine Function

Cosine, symbolized as \( \cos \), is another core trigonometric function that measures the ratio of the length of the adjacent side to the length of the hypotenuse, given a particular angle in a right-angled triangle.

It is defined by the formula: \( \cos(\theta) = \frac{\text{adjacent side}}{\text{hypotenuse}} \). In the case of our exercise, the angle \( t \) has its terminal side at the point \( (\sqrt{3}, -8) \), making \( \sqrt{3} \) the length of the side adjacent to \( t \). By calculating the hypotenuse using the Pythagorean theorem, we can then find \( \cos(t) \) as the x-coordinate divided by this hypotenuse. The positive value of \( \cos(t) \) indicates that the x-coordinate is positive, showing that the point is to the right of the y-axis even though it is in the third quadrant.

Exploring the Tangent Function

The tangent of an angle in trigonometry is a function that represents the ratio of the sine of the angle to the cosine of the angle. It can also be seen as the ratio of the length of the opposite side to the length of the adjacent side in a right-angled triangle.

The equation for tangent is given as: \( \tan(\theta) = \frac{\text{opposite side}}{\text{adjacent side}} = \frac{\sin(\theta)}{\cos(\theta)} \). In the exercise, to find \( \tan(t) \) for the angle \( t \) whose terminal side goes through the point \( (\sqrt{3}, -8) \) on the Cartesian plane, you divide the y-coordinate (-8) by the x-coordinate (\sqrt{3}). Here, the negative \( \tan(t) \) value indicates a downward slope of the line that forms angle \( t \) with the x-axis, consistent with an angle located in the third quadrant.

Understanding the tangent function is particularly useful for solving problems involving angles and slopes, such as those frequently encountered in physics and engineering.