Problem 3 - Basic Technical Mathematics with Calculus

Question

Find values of the trigonometric functions of the angle (in standard position) whose terminal side passes through the given points. For Exercises \(3-14,\) give answers in exact form. For Exercises 15 and \(16,\) the coordinates are approximate. $$(6,8)$$

Step-by-Step Solution

Answer

\( \sin(\theta) = \frac{4}{5}, \cos(\theta) = \frac{3}{5}, \tan(\theta) = \frac{4}{3} \).

Step 1: Understand the Problem

We need to find the sine, cosine, and tangent of the angle whose terminal side passes through the point (6,8). This means that the angle is in the standard position where the initial side is along the positive x-axis.

Step 2: Determine the Quadrant

Since both the x-coordinate (6) and the y-coordinate (8) are positive, the terminal side is in the first quadrant.

Step 3: Find the Hypotenuse

The point (6,8) and the origin (0,0) form a right triangle. Use the Pythagorean theorem to find the hypotenuse: \( r = \sqrt{x^2 + y^2} = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \).

Step 4: Calculate Sine of the Angle

The sine of the angle is the ratio of the opposite side to the hypotenuse: \( \sin(\theta) = \frac{y}{r} = \frac{8}{10} = \frac{4}{5} \).

Step 5: Calculate Cosine of the Angle

The cosine of the angle is the ratio of the adjacent side to the hypotenuse: \( \cos(\theta) = \frac{x}{r} = \frac{6}{10} = \frac{3}{5} \).

Step 6: Calculate Tangent of the Angle

The tangent of the angle is the ratio of the opposite side to the adjacent side: \( \tan(\theta) = \frac{y}{x} = \frac{8}{6} = \frac{4}{3} \).

Step 7: Verify the Results

Ensure that the calculated values are consistent with the conditions of the first quadrant, where both sine and cosine are positive.

Key Concepts

Pythagorean theorem

The Pythagorean theorem is a fundamental principle in geometry, especially useful when dealing with right triangles. It states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. The formula looks like this: \[a^2 + b^2 = c^2\]where \( a \) and \( b \) are the lengths of the two legs, and \( c \) is the hypotenuse.
When encountering a point like \((6, 8)\), this point describes the legs of a right triangle when connected back to the origin \((0, 0)\) on a coordinate plane. Here:

• \( x = 6 \) represents the horizontal leg
• \( y = 8 \) represents the vertical leg

These distances are used in the Pythagorean theorem to find the hypotenuse, \( r \):\[r = \sqrt{x^2 + y^2} = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10\]Understanding this concept is crucial. It not only allows us to find the hypotenuse but also aids in calculating trigonometric functions, which are valuable in many areas of mathematics and science.

right triangle

A right triangle is characterized by having one of its angles as a right angle, which is exactly \(90^\circ\). In the context of the problem, right triangles appear naturally when you draw a line from the origin to a point on the plane.
With the corners of the triangle positioned at the origin \((0,0)\), the x-coordinate \((6)\), and the y-coordinate \((8)\), the right angle is located at the origin:

• The leg along the x-axis: \(AB = 6\)
• The leg along the y-axis: \(BC = 8\)
• Hypotenuse formed: \(AC = 10\)

Right triangles are integral to understanding trigonometry because they allow you to apply trigonometric ratios such as sine, cosine, and tangent. For example, knowing the side measurements lets you find these ratios:

• Sine: \( \sin(\theta) = \frac{8}{10} = \frac{4}{5} \)
• Cosine: \( \cos(\theta) = \frac{6}{10} = \frac{3}{5} \)
• Tangent: \( \tan(\theta) = \frac{8}{6} = \frac{4}{3} \)

Remember, all these calculations were possible because we identified a right triangle.

standard position angle

An angle in standard position is one of the basics of trigonometry. It's an angle situated with its vertex at the origin of the coordinate plane. Here, the initial side of the angle is lined up along the positive x-axis. This is particularly important because this setup standardizes how angles are measured and understood, especially when calculating trigonometric functions.
For an angle in standard position:

• The rotation towards the terminal side determines the angle. Counter-clockwise is positive.
• Since the point \((6, 8)\) lies in the first quadrant, where both coordinates are positive, the angle's terminal side lies here as well.

In this first quadrant, trigonometric functions have the following characteristics:

• All trigonometric functions are positive.
• This provides a simplified environment to calculate the sine, cosine, and tangent values.

Such fundamental placements make the understanding of an angle's position crucial, as it heavily influences the characteristics of the trinomial functions associated with it.