Problem 4 - Algebra and Trigonometry

Question

a point on the terminal side of angle \(\theta\) is given. Find the exact value of each of the six trigonometric functions of \(\theta .\) $$ (3,7) $$

Step-by-Step Solution

Answer

The exact values of the six trigonometric functions are as follows: \(\sin(\theta) = \frac{7}{\sqrt{58}}\), \(\cos(\theta) = \frac{3}{\sqrt{58}}\), \(\tan(\theta) = \frac{7}{3}\), \(\csc(\theta) = \frac{\sqrt{58}}{7}\), \(\sec(\theta) = \frac{\sqrt{58}}{3}\), \(\cot(\theta) = \frac{3}{7}\).

Step 1: Identify the coordinates

Identify the given point’s coordinates. Here, the point’s coordinates are (3,7). In a coordinate plane, the x-coordinate is 3 (length along the x-axis) and y-coordinate is 7 (height along the y-axis). So, x=3 and y=7.

Step 2: Calculate the length of the hypotenuse

The length of the hypotenuse of the right triangle (r) can be calculated using the Pythagorean theorem: \(r = \sqrt{x^2 + y^2}\). Substituting x=3 and y=7, we get \(r = \sqrt{3^2 + 7^2} = \sqrt{9+49} = \sqrt{58}\).

Step 3: Find the trigonometric functions

Now, calculate each of the six trigonometric functions using their formulas against the lengths of sides: \n1. Sine function, \(\sin(\theta) = \frac{y}{r} = \frac{7}{\sqrt{58}}\), \n2. Cosine function, \(\cos(\theta) = \frac{x}{r} = \frac{3}{\sqrt{58}}\), \n3. Tangent function, \(\tan(\theta) = \frac{y}{x} = \frac{7}{3}\), \n4. Cosecant function, \(\csc(\theta) = \frac{r}{y} = \frac{\sqrt{58}}{7}\), \n5. Secant function, \(\sec(\theta) = \frac{r}{x} = \frac{\sqrt{58}}{3}\), \n6. Cotangent function, \(\cot(\theta) = \frac{x}{y} = \frac{3}{7}\).

Key Concepts

Understanding Right Triangles

A right triangle has one angle of 90 degrees, making it unique among triangles. In a right triangle, the side opposite the right angle is the hypotenuse, which is always the longest side. The other two sides are called the 'legs.'

• The hypotenuse is crucial because it's used to connect the concepts of geometry with trigonometry.
• It provides the diagonal distance from the right angle, influencing calculations in the Pythagorean theorem and trigonometric ratios.

In a coordinate plane, when a point like (3,7) is given, you can picture it forming a right triangle. The x-coordinate (3) and y-coordinate (7) form the triangle's legs. Understanding this connection helps us connect Cartesian coordinates with trigonometric concepts.

Exploring the Pythagorean Theorem

The Pythagorean theorem is a fundamental principle that relates the sides of a right triangle. It states that for a right triangle, the square of the hypotenuse (\(c\)) is equal to the sum of the squares of the other two sides (\(a\) and \(b\)): \[ a^2 + b^2 = c^2 \].

• This theorem helps us find the hypotenuse length when the other two sides are known.
• In our exercise, this was used to calculate \(r\), or the distance from the origin to the point (3,7).
• For \(x = 3\) and \(y = 7\), it becomes \( 3^2 + 7^2 = r^2 \) resulting in \( r = \sqrt{58} \).

By applying this theorem, we can easily determine the hypotenuse and use it to find tignometric functions.

Navigating the Coordinate Plane

The coordinate plane consists of two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). It is used to locate points in a two-dimensional space, like (3,7).

• Points are identified by coordinates (x, y), with 'x' indicating horizontal position and 'y' indicating vertical position.
• The origin (0,0) is where the axes intersect, serving as a reference point.
• By plotting points, you can visualize geometric shapes and calculate distances between them.

Understanding the coordinate plane is key in trigonometry, as it allows for the graphical interpretation of angles and distances. This makes visualizing the formation of right triangles from points and angles intuitive, bridging algebra and geometry.

Sailing Through the Sine Function

The sine function is one of the primary trigonometric functions and is fundamental to understanding triangles and angles. In a right triangle, the sine function of an angle \(\theta\) is defined as the ratio of the length of the side opposite the angle to the hypotenuse: \(\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}\).

• From our example, it is calculated as \(\sin(\theta) = \frac{7}{\sqrt{58}}\) with 7 being the y-coordinate.
• This function helps in determining height or vertical distance in applications.
• It is extensively used in waves, oscillations, and circles, due to its periodic nature.

Understanding sine, along with other trigonometric functions, expands our ability to solve real-world problems involving angles, from navigation to designing complex geometric shapes.