Problem 7
Question
a point on the terminal side of angle \(\theta\) is given. Find the exact value of each of the six trigonometric functions of \(\theta .\) $$ (-2,-5) $$
Step-by-Step Solution
Verified Answer
The six trigonometric functions for the angle \( \theta \) are: \( sin(\theta) = -5 / \sqrt{29} \), \( cos(\theta) = -2 / \sqrt{29} \), \( tan(\theta) = 2.5 \), \( csc(\theta) = \sqrt{29} / -5 \), \( sec(\theta) = \sqrt{29} / -2 \), and \( cot(\theta) = 0.4 \).
1Step 1: Identify the coordinates
The terminal point is (-2,-5) which is the point on the unit circle corresponding to the angle \( \theta \). Here, X-coordinate corresponds to -2 and Y-coordinate corresponds to -5.
2Step 2: Calculate the radius using Pythagorean theorem
The radius, or the distance from the origin (0,0) to the point (-2,-5) is calculated using Pythagorean theorem. It is given by \( r = \sqrt{x^2 + y^2} = \sqrt{(-2)^2 + (-5)^2} = \sqrt{29} \)
3Step 3: Calculate Trigonometric function values
Each of the six trigonometric functions can be found by appropriate ratios:i) The sine function \( sin(\theta) = y / r = -5 / \sqrt{29} \), ii) The cosine function \( cos(\theta) = x / r = -2 / \sqrt{29} \),iii) The tangent function \( tan(\theta) = y / x = -5 / -2 = 5/2 = 2.5 \),iv) The cosecant function \( csc(\theta) \) is the reciprocal of \( sin(\theta) \), hence \( csc(\theta) = r / y = \sqrt{29} / -5 \),v) The secant function \( sec(\theta) \) is the reciprocal of \( cos(\theta) \), hence \( sec(\theta) = r / x = \sqrt{29} / -2 \),vi) The cotangent function \( cot(\theta) \) is the reciprocal of \( tan(\theta) \), hence \( cot(\theta) = x / y = -2 / -5 = 2/5 = 0.4 \).
Key Concepts
Sine FunctionCosine FunctionTangent FunctionCosecant FunctionCotangent Function
Sine Function
The sine function is a fundamental trigonometric function that relates the angle of a right triangle to the ratio of the opposite side to the hypotenuse. It is defined for an angle \( \theta \) as \( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \).
When dealing with coordinate systems, this translates to \( \sin(\theta) = \frac{y}{r} \), where \( y \) is the vertical coordinate of the point and \( r \) is the distance from the origin to the point.
In this context, \( r \) acts as the hypotenuse.
In our example, with point (-2,-5), the sine of the angle \( \theta \) is \( \sin(\theta) = \frac{-5}{\sqrt{29}} \). This negative value indicates the point is below the x-axis in the Cartesian plane.
When dealing with coordinate systems, this translates to \( \sin(\theta) = \frac{y}{r} \), where \( y \) is the vertical coordinate of the point and \( r \) is the distance from the origin to the point.
In this context, \( r \) acts as the hypotenuse.
In our example, with point (-2,-5), the sine of the angle \( \theta \) is \( \sin(\theta) = \frac{-5}{\sqrt{29}} \). This negative value indicates the point is below the x-axis in the Cartesian plane.
Cosine Function
The cosine function is another primary trigonometric function. It connects the adjacent side of a right triangle to its hypotenuse. It's given by \( \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \).
In a coordinate system, this translates into \( \cos(\theta) = \frac{x}{r} \).
This means that \( x \) is the horizontal coordinate of the point on the plane.
For the point (-2, -5), to find \( \cos(\theta) \), we use \( \cos(\theta) = \frac{-2}{\sqrt{29}} \).
The negative result signifies that the point lies to the left of the y-axis.
In a coordinate system, this translates into \( \cos(\theta) = \frac{x}{r} \).
This means that \( x \) is the horizontal coordinate of the point on the plane.
For the point (-2, -5), to find \( \cos(\theta) \), we use \( \cos(\theta) = \frac{-2}{\sqrt{29}} \).
The negative result signifies that the point lies to the left of the y-axis.
Tangent Function
The tangent function is a trigonometric function that describes the ratio of the sine function to the cosine function. It can alternatively be defined as the ratio of the opposite side to the adjacent side in a right triangle. In formulas, this is \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \) or \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \).
When using coordinates, it is given by \( \tan(\theta) = \frac{y}{x} \).
From our point (-2, -5), \( \tan(\theta) = \frac{-5}{-2} = 2.5 \), which simplifies directly to 2.5.
This value, being positive, reveals that both sine and cosine are of the same sign, indicating a quadrant where the tangent is positive.
When using coordinates, it is given by \( \tan(\theta) = \frac{y}{x} \).
From our point (-2, -5), \( \tan(\theta) = \frac{-5}{-2} = 2.5 \), which simplifies directly to 2.5.
This value, being positive, reveals that both sine and cosine are of the same sign, indicating a quadrant where the tangent is positive.
Cosecant Function
Cosecant, or \( \csc(\theta) \), is the reciprocal of the sine function. It is defined as \( \csc(\theta) = \frac{1}{\sin(\theta)} \) or, using coordinates, \( \csc(\theta) = \frac{r}{y} \).
This function can sometimes be more convenient when expressing reciprocals in calculations.
For our exercise, using point (-2, -5), we find \( \csc(\theta) = \frac{\sqrt{29}}{-5} \).
The negative value reflects the direction of \( y \), pointing downward from the origin.
This function can sometimes be more convenient when expressing reciprocals in calculations.
For our exercise, using point (-2, -5), we find \( \csc(\theta) = \frac{\sqrt{29}}{-5} \).
The negative value reflects the direction of \( y \), pointing downward from the origin.
Cotangent Function
The cotangent function is the reciprocal of the tangent function. It can be represented as \( \cot(\theta) = \frac{1}{\tan(\theta)} \) or simply as \( \cot(\theta) = \frac{x}{y} \) when using coordinates.
It simplifies many trigonometric calculations involving angles.
In our example problem with the point (-2, -5), \( \cot(\theta) = \frac{-2}{-5} = 0.4 \).
This positive result suggests both \( x \) and \( y \) are negative, aligning with the quadrant characteristics where cotangent is positive.
It simplifies many trigonometric calculations involving angles.
In our example problem with the point (-2, -5), \( \cot(\theta) = \frac{-2}{-5} = 0.4 \).
This positive result suggests both \( x \) and \( y \) are negative, aligning with the quadrant characteristics where cotangent is positive.
Other exercises in this chapter
Problem 7
In Exercises 5–12, graph two periods of the given tangent function. $$ y=\frac{1}{2} \tan 2 x $$
View solution Problem 7
Determine the amplitude and period of each function. Then graph one period of the function. $$y=\sin 2 x$$
View solution Problem 7
In Exercises \(7-12,\) find the radian measure of the central angle of a circle of radius \(r\) that intercepts an arc of length \(s\). $$ Radius, r \quad Arc L
View solution Problem 8
Find the exact value of each expression. $$ \cos ^{-1} \frac{\sqrt{2}}{2} $$
View solution