Problem 1 - Contemporary Precalculus

Question

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

Step-by-Step Solution

Answer

Question: Calculate the sine, cosine, and tangent of the angle in standard position whose terminal side contains the point (2,3). Answer: The sine, cosine, and tangent of the angle are as follows: $$\sin(\theta) = \frac{3}{\sqrt{13}}$$ $$\cos(\theta) = \frac{2}{\sqrt{13}}$$ $$\tan(\theta) = \frac{3}{2}$$

Step 1: Calculate the hypotenuse

The given point is P(2,3). To find the length of the hypotenuse, we will use the Pythagorean theorem: \(h = \sqrt{x^2 + y^2}\). In this case, the x-coordinate is 2, and the y-coordinate is 3. $$h = \sqrt{2^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13}$$

Step 2: Determine the angle measure

We'll use the tangent function to find the angle from the positive x-axis (also called reference angle). Tangent is calculated by dividing the y-coordinate by the x-coordinate (\(\tan(\theta) = \frac{y}{x}\)). So, $$\tan(\theta) = \frac{3}{2}$$ To find the angle measure, we will use the arctangent function: \(\theta = \arctan (\frac{3}{2})\). Calculating the value of \(\theta\) gives: $$\theta \approx 56.31^{\circ}$$

Step 3: Calculate the sine function

Now that we have the length of the hypotenuse and the angle measure, we can find the sine of the angle by dividing the y-coordinate (opposite side) by the length of the hypotenuse (\(\sin(\theta) = \frac{y}{h}\)). $$\sin(\theta) = \frac{3}{\sqrt{13}}$$

Step 4: Calculate the cosine function

To find the cosine of the angle, divide the x-coordinate (adjacent side) by the length of the hypotenuse (\(\cos(\theta) = \frac{x}{h}\)). $$\cos(\theta) = \frac{2}{\sqrt{13}}$$

Step 5: Calculate the tangent function

Finally, to compute the tangent of the angle, divide the y-coordinate (opposite side) by the x-coordinate (adjacent side) (\(\tan(\theta) = \frac{y}{x}\)). $$\tan(\theta) = \frac{3}{2}$$ So, the trigonometric functions are as follows: $$\sin(\theta) = \frac{3}{\sqrt{13}}$$ $$\cos(\theta) = \frac{2}{\sqrt{13}}$$ $$\tan(\theta) = \frac{3}{2}$$

Key Concepts

Pythagorean Theorem

When it comes to understanding right triangles, the Pythagorean theorem is an essential tool. It states that for any 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 other two sides. This theorem can be expressed with the formula: \[h^2 = a^2 + b^2\] where \(h\) is the hypotenuse, and \(a\) and \(b\) are the other two sides. In the context of trigonometric functions, we often start with a point on the Cartesian coordinate system, like \( (2,3) \), and use these coordinates as the legs of a right triangle, with the origin (0,0) as one vertex. Here we use the theorem to calculate the hypotenuse by plugging in the coordinates: \[h = \sqrt{2^2 + 3^2} = \sqrt{13}\] which becomes the reference for evaluating trigonometric ratios.

Tangent Function

The tangent function is a fundamental trigonometric function that relates an angle in a right-angled triangle to the ratio of its opposite side to the adjacent side. Mathematically, it's described as \[\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\] where \(\theta\) is the angle in question. If we're looking at a coordinate system and referring to a point \( (x, y) \) representing the sides of the triangle, the tangent function can tell us the slope of the line that connects that point to the origin. In our exercise: \[\tan(\theta) = \frac{y}{x} = \frac{3}{2}\] This function is particularly useful when trying to find the angle \(\theta\), which can be done by using the arc tangent function often represented as \(\arctan\) or \(\tan^{-1}\). The value of \(\theta\) helps us to determine the correct angle that corresponds to the given tangent ratio.

Sine and Cosine

The sine and cosine functions are two of the most important trigonometric functions, deeply connected to the concept of an angle in a circle or triangle.

For an angle \( \theta \) in standard position, the sine function \( \sin(\theta) \) represents the ratio of the length of the opposite side to the hypotenuse, while the cosine function \( \cos(\theta) \) gives the ratio of the length of the adjacent side to the hypotenuse. Their formulas are: \[\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}\] and \[\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}\] respectively. In the given exercise, using the point \( (2,3) \) and the hypotenuse \(\sqrt{13}\), we would calculate them as: \[\sin(\theta) = \frac{3}{\sqrt{13}}\] and \[\cos(\theta) = \frac{2}{\sqrt{13}}\] These ratios are crucial because they allow us to evaluate the sine and cosine for any angle, providing the foundation for solving various problems in mathematics, physics, engineering, and even computer science.