Problem 5 - Contemporary Precalculus

Question

Evaluate the trigonometric functions at the angle (in standard position) whose terminal side contains the given point. $$(-3,-\sqrt{2})$$

Step-by-Step Solution

Answer

Question: Given the point (-3, -√2), find the sine, cosine, and tangent of the angle in standard position whose terminal side contains the given point. Answer: Given the point (-3, -√2), we have calculated the values of the trigonometric functions as follows: $$\sin{\theta} = -\frac{\sqrt{2}}{\sqrt{11}}$$ $$\cos{\theta} = -\frac{3}{\sqrt{11}}$$ $$\tan{\theta} = \frac{\sqrt{2}}{3}$$

Step 1: 1. Determine the angle in standard position

: First, let's plot the point (-3, -√2) on the Cartesian plane. Since it is in the third quadrant (both x and y are negative), the angle in standard position will be in the range between \(180^\circ\) and \(270^\circ\) (or between \(\pi\) and \(\dfrac{3\pi}{2}\) in radians). Now, let's find the reference angle by using the inverse tangent function (typically denoted as atan or tan\(^{-1}\)): $$\theta_{ref} = \tan^{-1}\left( \frac{y}{x} \right) = \tan^{-1}\left( \frac{-\sqrt{2}}{-3} \right) = \tan^{-1}\left(\frac{\sqrt{2}}{3}\right)$$ After obtaining the reference angle, we can get the angle in standard position in radians by adding \(\pi\) to the reference angle: $$\theta = \theta_{ref} + \pi$$

Step 2: 2. Calculate the radius (r)

: To calculate the trigonometric functions, we need to find the radius (r) of the triangle formed by the given point and the origin. Using the Pythagorean theorem: $$r = \sqrt{x^2 + y^2} = \sqrt{(-3)^2 + (-\sqrt{2})^2} = \sqrt{9+2} = \sqrt{11}$$

Step 3: 3. Evaluate trigonometric functions

: Now that we have the angle and the radius, we can easily evaluate the sine, cosine, and tangent using the point coordinates. 1. Sine function: $$\sin{\theta} = \frac{y}{r} = \frac{-\sqrt{2}}{\sqrt{11}} = -\frac{\sqrt{2}}{\sqrt{11}}$$ 2. Cosine function: $$\cos{\theta} = \frac{x}{r} = \frac{-3}{\sqrt{11}} = -\frac{3}{\sqrt{11}}$$ 3. Tangent function: $$\tan{\theta} = \frac{\sin{\theta}}{\cos{\theta}} = \frac{-\frac{\sqrt{2}}{\sqrt{11}}}{-\frac{3}{\sqrt{11}}} = \frac{\sqrt{2}}{3}$$ So, the values of the trigonometric functions at the angle in standard position with the given point are: $$\sin{\theta} = -\frac{\sqrt{2}}{\sqrt{11}}$$ $$\cos{\theta} = -\frac{3}{\sqrt{11}}$$ $$\tan{\theta} = \frac{\sqrt{2}}{3}$$

Key Concepts

Trigonometry

Trigonometry is a branch of mathematics that studies the relationships between the angles and sides of triangles. It is essential for solving problems that involve right triangles, but it also extends to more complex geometric shapes and even circles.
Trigonometric functions, like sine (\(\sin\theta\)), cosine (\(\cos\theta\)), and tangent (\(\tan\theta\)), relate the angles of triangles to the ratios of their sides. These functions are fundamental in calculating angles and distances in various scientific and engineering contexts.
Trigonometry is not limited to right angles; it also plays a crucial role in analyzing periodic phenomena, such as sound waves and light waves, thanks to its oscillatory nature.

• Sine (\(\sin\theta\)): Ratio of the opposite side to the hypotenuse.
• Cosine (\(\cos\theta\)): Ratio of the adjacent side to the hypotenuse.
• Tangent (\(\tan\theta\)): Ratio of the opposite side to the adjacent side.

Cartesian coordinates

The Cartesian coordinate system is a two-dimensional plane defined by two perpendicular axes: the x-axis and the y-axis. It allows us to locate points uniquely using ordered pairs \( (x, y) \). This system is an essential framework for graphing equations and analyzing geometric properties.
In trigonometry, we often refer to points in the plane, which helps us determine various properties of angles and triangles. A point such as \((-3, -\sqrt{2})\) lies in the third quadrant where both x and y are negative. Each quadrant helps us understand the angle formations and trigonometric sign conventions necessary for calculations.

• First Quadrant: \(x > 0, y > 0\)
• Second Quadrant: \(x < 0, y > 0\)
• Third Quadrant: \(x < 0, y < 0\)
• Fourth Quadrant: \(x > 0, y < 0\)

Reference Angle

A reference angle is the minimum angle a given angle makes with the x-axis. It is always positive and lies between \(0\) and \(\frac{\pi}{2}\) radians, or \(0\) to \(90^\circ\). Reference angles are crucial as they provide a simplified way to evaluate trigonometric functions for angles located in different quadrants.
Using the arctangent function helps in finding the reference angle from a point. For example, for the point \((-3, -\sqrt{2})\), the reference angle is derived as \(\tan^{-1}\left(\frac{\sqrt{2}}{3}\right)\).
Essentially, it gives the acute angle equivalent and simplifies the understanding of trigonometric values in any quadrant. The concept leverages symmetrical properties of sine, cosine, and tangent functions.

Standard Position Angle

An angle in standard position is located with its vertex at the origin and its initial side along the positive x-axis. A critical concept in trigonometry, it helps to universally define angles around the unit circle.
When evaluating points such as \((-3, -\sqrt{2})\), the angle is formed by rotating counterclockwise from the positive x-axis to the terminal side that contains the point.
In this context, the standard position angle is calculated by adding \(\pi\) (or \(180^\circ\)) to the reference angle for points located in the third quadrant. This adjustment ensures that the angle reflects the correct rotational measure necessary for computing trigonometric functions.

• Third Quadrant: Adds \(\pi\) to the reference angle.
• Angles cover \(0\) to \(2\pi\) in radians or \(0^\circ\) to \(360^\circ\)