Problem 11 - 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. $$(7,7)$$

Step-by-Step Solution

Answer

The trigonometric functions are \( \sin \theta = \frac{\sqrt{2}}{2} \), \( \cos \theta = \frac{\sqrt{2}}{2} \), \( \tan \theta = 1 \), \( \csc \theta = \sqrt{2} \), \( \sec \theta = \sqrt{2} \), and \( \cot \theta = 1 \).

Step 1: Calculate the radius (r)

To find the trigonometric functions, we first need to find the radius \( r \) of the circle that passes through the point \((7, 7)\). The radius is given by the distance from the origin to the point, calculated using the Pythagorean theorem:\[r = \sqrt{x^2 + y^2} = \sqrt{7^2 + 7^2}\]Calculating this gives:\[r = \sqrt{49 + 49} = \sqrt{98} = 7\sqrt{2}\]

Step 2: Calculate Sine \((\sin \theta)\)

The sine of the angle \( \theta \) is given by:\[\sin \theta = \frac{y}{r}\]Substitute \( y = 7 \) and \( r = 7\sqrt{2} \):\[\sin \theta = \frac{7}{7\sqrt{2}} = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}\]

Step 3: Calculate Cosine \((\cos \theta)\)

The cosine of the angle \( \theta \) is given by:\[\cos \theta = \frac{x}{r}\]Substitute \( x = 7 \) and \( r = 7\sqrt{2} \):\[\cos \theta = \frac{7}{7\sqrt{2}} = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}\]

Step 4: Calculate Tangent \((\tan \theta)\)

The tangent of the angle \( \theta \) is the ratio of the sine and cosine:\[\tan \theta = \frac{y}{x} = \frac{7}{7} = 1\]

Step 5: Calculate Cosecant \((\csc \theta)\)

The cosecant of \( \theta \) is the reciprocal of sine:\[\csc \theta = \frac{1}{\sin \theta} = \frac{1}{\frac{\sqrt{2}}{2}} = \sqrt{2}\]

Step 6: Calculate Secant \((\sec \theta)\)

The secant of \( \theta \) is the reciprocal of cosine:\[\sec \theta = \frac{1}{\cos \theta} = \frac{1}{\frac{\sqrt{2}}{2}} = \sqrt{2}\]

Step 7: Calculate Cotangent \((\cot \theta)\)

The cotangent of \( \theta \) is the reciprocal of tangent:\[\cot \theta = \frac{1}{\tan \theta} = \frac{1}{1} = 1\]

Key Concepts

sine

Sine is a fundamental trigonometric function that describes the relationship between the opposite side of a right-angled triangle and its hypotenuse. When dealing with an angle in standard position, where the vertex is at the origin and the initial side is along the positive x-axis, the sine of the angle is defined as the ratio of the y-coordinate of a point on the circle to the radius of the circle.

Here's how to grasp it with the example of the point \(7,7\):

• The y-coordinate is 7.
• The radius of the circle, using the Pythagorean theorem, is \(7\sqrt{2}\).

Thus, the sine of the angle is calculated as \( \sin \theta = \frac{y}{r} = \frac{7}{7\sqrt{2}} = \frac{\sqrt{2}}{2} \).

This concept helps us understand how a vertical position relates to the circle's radius. Therefore, sine values can range from -1 to 1, indicating the maximum and minimum positions on the unit circle.

cosine

Cosine is another critical trigonometric function. It measures the ratio of the adjacent side of a right triangle to its hypotenuse. For an angle in standard position, the cosine is the ratio of the x-coordinate of a point on the circle to the radius.

Let's break it down with the point \(7,7\):

• The x-coordinate is 7.
• The radius is again \(7\sqrt{2}\).

The cosine is then calculated as \( \cos \theta = \frac{x}{r} = \frac{7}{7\sqrt{2}} = \frac{\sqrt{2}}{2} \).

Cosine represents the horizontal aspect of the angle on the unit circle. Its values vary between -1 and 1, indicating how far left or right the point is from the origin. Understanding cosine lets us comprehend an angle's horizontal component effectively.

tangent

Tangent is a trigonometric function connecting sine and cosine, defined as the ratio of sine to cosine. It explains the slope of the line formed from the origin to a point on the circle. The tangent is calculated as the ratio of the y-coordinate to the x-coordinate.

Considering \(7,7\):

• Both coordinates are equal, y = x = 7.

Thus, \( \tan \theta = \frac{y}{x} = \frac{7}{7} = 1 \).

This value indicates a perfect 45-degree angle (or \( \frac{\pi}{4} \) radians) relative to the x-axis. Tangent can take any real number value and gives insight into the steepness or angle of inclination, making it useful in calculating real-world slopes and angles.