Question

The terminal side of an angle in standard position passes through the given point. Sketch the angle, compute the distance \(r\) from the origin to the point, and write the six trigonometric functions of the angle. Work to three significant digits. $$(-5.13,-1.17)$$

Short answer

The distance r is approximately 5.27. The six trigonometric functions (to three significant digits) are: sin: -0.222, cos: -0.973, tan: 0.228, csc: -4.50, sec: -1.03, cot: 4.39.

Step-by-step solution

Sketch the Angle

Plot the point (-5.13,-1.17) on the coordinate plane. Draw the terminal side of the angle from the origin (0,0) through the point, and the initial side along the positive x-axis. This creates a right triangle with the x-axis, where the angle in question is located in the third quadrant.

Compute the Distance r

Use the distance formula, or Pythagorean Theorem, to find the distance between the origin and the point. The distance r is the hypotenuse of the right triangle and can be computed with the formula \(r = \sqrt{x^2 + y^2}\), where x and y are the coordinates of the point. Therefore: \(r = \sqrt{(-5.13)^2 + (-1.17)^2} = \sqrt{26.3169 + 1.3689} = \sqrt{27.6858}\).

Write the Six Trigonometric Functions

Compute the six trigonometric functions using the coordinates of the point and the computed distance r. Since the point is in the third quadrant, both sine and cosine are negative.\ The six trigonometric functions are: \ Sine (sin): \(\sin(\theta) = \frac{y}{r}\)\ Cosine (cos): \(\cos(\theta) = \frac{x}{r}\)\ Tangent (tan): \(\tan(\theta) = \frac{y}{x}\)\ Cosecant (csc): \(\csc(\theta) = \frac{r}{y}\)\ Secant (sec): \(\sec(\theta) = \frac{r}{x}\)\ Cotangent (cot): \(\cot(\theta) = \frac{x}{y}\)\ Substitute the values of x, y, and r (rounded to three significant digits):\ \(\sin(\theta) = \frac{-1.17}{5.27}\)\ \(\cos(\theta) = \frac{-5.13}{5.27}\)\ \(\tan(\theta) = \frac{-1.17}{-5.13}\)\ \(\csc(\theta) = \frac{5.27}{-1.17}\)\ \(\sec(\theta) = \frac{5.27}{-5.13}\)\ \(\cot(\theta) = \frac{-5.13}{-1.17}\)

Key concepts

Standard Position of an Angle

Understanding the standard position of an angle is crucial in trigonometry. An angle is said to be in standard position when its vertex is at the origin of the coordinate plane and its initial side lies along the positive x-axis. The terminal side of the angle is the other ray that moves counterclockwise from the initial side to form the angle.

For the point (-5.13, -1.17), envisioning the angle starts by plotting the point on the Cartesian coordinate plane. Then, we draw a ray from the origin through this point to represent the terminal side of the angle. The angle in this exercise is in the third quadrant, implying it's between 180° and 270°. This helps us understand the nature of the trigonometric functions for the angle as they will be influenced by the quadrant in which the angle resides.

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in geometry that relates the sides of a right triangle. According to the theorem, the square of the length of the hypotenuse (\the longest side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. Presented as an equation, it looks like this: \(a^2 + b^2 = c^2\), where 'c' represents the hypotenuse.

In our exercise, the distance from the origin to point (-5.13, -1.17) can be found using this theorem by considering our x and y coordinates as the two shorter sides of a right triangle, and the distance 'r' as the hypotenuse. Therefore, by substituting \(a = -5.13\) and \(b = -1.17\), we can solve for 'r'. This method enables students to calculate the length of 'r' accurately, which is a key step in determining the six trigonometric functions.

Coordinates and Trigonometry

When dealing with trigonometry in the context of coordinates, we apply the trigonometric ratios sine, cosine, and tangent to a right-angled triangle formed by the x and y axes and a terminal side passing through a certain point.

In our case, the coordinate point (-5.13, -1.17) gives us the lengths of the sides relative to the angle located at the origin. These lengths are negative because the point lies in the third quadrant. Using these coordinates, the trigonometric functions can be calculated as the ratios of the sides. For example, the sine function is the ratio of the y-coordinate to the distance 'r', which is the hypotenuse in our right triangle scenario. This allows us to understand the signs and values of the trigonometric functions based on the coordinates of the point.

Trigonometry in the Coordinate Plane

When we place trigonometry in the setting of the coordinate plane, we open up the ability to analyze angles and triangles in a grid that extends infinitely in all directions. The process of plotting points, such as (-5.13, -1.17), provides a visual representation of the trigonometric functions as ratios that can change based on the location of the point relative to the quadrants.

Each quadrant has its own 'personality' regarding the signs of the trigonometric functions. Since our point is in the third quadrant, both the sine and cosine functions are negative. This reflects the fact that in the third quadrant, the x and y coordinates of any point are both negative, as the point lies below the x-axis and to the left of the y-axis. The process of calculating the six trigonometric functions is a practice in applying these ratios to the coordinates and the derived length 'r' to solve for each function's value.