Problem 15 - Contemporary Precalculus

Question

In Exercises \(7-16,\) evaluate all six trigonometric finctions at \(t\) where the given point lies on the terminal side of an angle of \(t\) radians in standard position. $$(1+\sqrt{2}, 3)$$

Step-by-Step Solution

Answer

Question: Given the point (1+√2, 3) on the terminal side of an angle t, find the values of all six trigonometric functions at angle t. Answer: For angle t, the six trigonometric functions are: sin(t) = 3/(√(12+2√2)), cos(t) = (1+√2)/(√(12+2√2)), tan(t) = 3/(1+√2), csc(t) = (√(12+2√2))/3, sec(t) = (√(12+2√2))/(1+√2), and cot(t) = (1+√2)/3.

Step 1: Identify the point in Cartesian coordinates

The point on the terminal side of angle \(t\) is given as \((1+\sqrt{2}, 3)\). Therefore, this point has coordinates \(x = 1+\sqrt{2}\) and \(y = 3\).

Step 2: Find the hypotenuse (radius) using the Pythagorean theorem

The hypotenuse (radius) is the distance between the origin \((0,0)\) and the point \((x,y)\). We can use the Pythagorean theorem to find that distance: \(r = \sqrt{x^2 + y^2}\). In our case, $$r = \sqrt{(1+\sqrt{2})^2 + 3^2} = \sqrt{(1+2\sqrt{2}+2)+9} = \sqrt{12+2\sqrt{2}}.$$

Step 3: Evaluate the trigonometric functions

Now we can use the definitions of trigonometric functions based on the values of \(x\), \(y\), and \(r\), which we found in steps 1 and 2. 1. Sine function: \(\sin(t) = \frac{y}{r}=\frac{3}{\sqrt{12+2\sqrt{2}}}\). 2. Cosine function: \(\cos(t) = \frac{x}{r} = \frac{1+\sqrt{2}}{\sqrt{12+2\sqrt{2}}}\). 3. Tangent function: \(\tan(t) = \frac{y}{x} = \frac{3}{1+\sqrt{2}}\). 4. Cosecant function: \(\csc(t) = \frac{1}{\sin(t)}= \frac{\sqrt{12+2\sqrt{2}}}{3}\). 5. Secant function: \(\sec(t) = \frac{1}{\cos(t)} = \frac{\sqrt{12+2\sqrt{2}}}{1+\sqrt{2}}\). 6. Cotangent function: \(\cot(t) = \frac{1}{\tan(t)} = \frac{1+\sqrt{2}}{3}\). The values of the six trigonometric functions at \(t\) radians are: $$ \sin(t) = \frac{3}{\sqrt{12+2\sqrt{2}}}, \cos(t) = \frac{1+\sqrt{2}}{\sqrt{12+2\sqrt{2}}}, \tan(t) = \frac{3}{1+\sqrt{2}}, \csc(t) = \frac{\sqrt{12+2\sqrt{2}}}{3}, \sec(t) = \frac{\sqrt{12+2\sqrt{2}}}{1+\sqrt{2}},\cot(t) = \frac{1+\sqrt{2}}{3}. $$

Key Concepts

Pythagorean theorem

The Pythagorean theorem is a fundamental principle in geometry that describes a special relationship between the sides of a right-angled triangle. It states that 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. Expressed mathematically, if we have a triangle with sides of length 'a', 'b', and hypotenuse 'c', the theorem is written as: \[c^2 = a^2 + b^2\]

The theorem is not only essential for various geometric calculations but also crucial in trigonometry, especially when evaluating trigonometric functions. In our exercise, we used it to find the length of the hypotenuse, or radius ‘r’, which is the distance from the origin to a point \(x, y\) in the Cartesian coordinate system. By applying the Pythagorean theorem, we calculated 'r' as the square root of the sum of the squares of the given coordinates, a step which is the foundation for determining the values of trigonometric functions. This concept is particularly important because it helps establish a link between geometric shapes and trigonometric ratios, allowing us to solve many practical problems in fields such as engineering, physics, and even computer science.

Radians in standard position

Radians are a measure of angle based on the radius of a circle. One radian is the angle created when the arc length is equal to the radius of the circle. When we discuss angles in standard position, we mean that the angle's vertex is at the origin (0,0) of a coordinate system, and one ray (the initial side) lies along the positive x-axis. The second ray (the terminal side) sweeps out the angle and can lie in any of the four quadrants.

Using radians is especially important in trigonometry because it provides a natural way of relating angular motion to linear motion, such as in the case of circular waves or rotations. In the given exercise, the point \(1+\sqrt{2}, 3\) lies on the terminal side of an angle of 't' radians in standard position. Understanding the standard position is crucial because it sets the stage for evaluating trigonometric functions accurately. It ensures that we analyze the angle and its trigonometric ratios within a consistent and universally applicable framework. This concept is essential for anyone learning trigonometry, as it serves as a gateway to a wide range of problems that can be solved using trigonometric functions.

Evaluating trigonometric functions

Evaluating trigonometric functions involves finding the values of functions like sine, cosine, and tangent for a given angle. These functions help describe the relationships between the angles and sides of triangles, especially in a right-angled triangle. In the context of the exercise, these functions are evaluated for a specific angle 't' in radians, which determines a point on a terminal side in a standard position.

Let's quickly go over the main trigonometric functions:

• Sine (sin): ratio of the opposite side to the hypotenuse.
• Cosine (cos): ratio of the adjacent side to the hypotenuse.
• Tangent (tan): ratio of the opposite side to the adjacent side, or the sine over the cosine.

There are also the reciprocal functions:

• Cosecant (csc), which is the reciprocal of sine.
• Secant (sec), which is the reciprocal of cosine.
• Cotangent (cot), which is the reciprocal of tangent.

By knowing the coordinates of a point on the terminal side and the hypotenuse obtained using the Pythagorean theorem, we can calculate the trigonometric function values. This process is critical in various applications, such as physics simulations, signal processing, and even in the analysis of periodic phenomena. A clear understanding of how to evaluate these functions at any given angle is vital for students embarking on careers in STEM fields.