Resourcesdown chevron
Testimonials
Plans
Sign in
Sign up
Resourcesright arrow
Testimonials
Plans
Bytelearn - cat image with glassesAI tutor

Welcome to Bytelearn!

Let’s check out your problem:

Sketch an angle theta in standard position such that theta has the least possible positive measure, and the point (-7,-24) is on the terminal side of theta. Then find the values of the six trigonometric functions for the angle. Rationalize denominators when applicable. Do not use a calculator.

Sketch an angle θ \theta in standard position such that θ \theta has the least possible positive measure, and the point (−7,−24) (-7,-24) is on the terminal side of θ \theta . Then find the values of the six trigonometric functions for the angle. Rationalize denominators when applicable. Do not use a calculator.

View step-by-step helpView step-by-step help
Homeright chevron icon
Math Problemsright chevron icon
Algebra 2right chevron icon
Find the constant of variationright chevron icon

Full solution

Q. Sketch an angle θ \theta in standard position such that θ \theta has the least possible positive measure, and the point (−7,−24) (-7,-24) is on the terminal side of θ \theta . Then find the values of the six trigonometric functions for the angle. Rationalize denominators when applicable. Do not use a calculator.
  1. Determine Quadrant: First, we need to determine the quadrant in which the angle θ\theta's terminal side lies since the point (−7,−24)(-7, -24) has both negative xx and yy coordinates, θ\theta is in the third quadrant.
  2. Find Reference Angle: To find the reference angle, we use the coordinates (−7,−24)(-7, -24). The reference angle is calculated using the tangent, which is the ratio of the y-coordinate to the x-coordinate. So, tan⁡(reference angle)=247\tan(\text{reference angle}) = \frac{24}{7}.
  3. Calculate Trigonometric Functions: Now, we calculate the six trigonometric functions for θ\theta. Since θ\theta is in the third quadrant, sine and cosine are negative, and tangent is positive. We start by finding the hypotenuse using the Pythagorean theorem: hypotenuse=(−7)2+(−24)2=49+576=625=25\text{hypotenuse} = \sqrt{(-7)^2 + (-24)^2} = \sqrt{49 + 576} = \sqrt{625} = 25.
  4. Find Hypotenuse: The sine of theta is the y-coordinate divided by the hypotenuse, so sin⁡(θ)=−2425\sin(\theta) = -\frac{24}{25}. The cosine of theta is the x-coordinate divided by the hypotenuse, so cos⁡(θ)=−725\cos(\theta) = -\frac{7}{25}. The tangent of theta is the y-coordinate divided by the x-coordinate, so tan⁡(θ)=247\tan(\theta) = \frac{24}{7}.
  5. Reciprocal Trigonometric Functions: For the reciprocal trigonometric functions, cosecant is the reciprocal of sine, so csc⁡(θ)=−2524\csc(\theta) = -\frac{25}{24}. Secant is the reciprocal of cosine, so sec⁡(θ)=−257\sec(\theta) = -\frac{25}{7}. Cotangent is the reciprocal of tangent, so cot⁡(θ)=724\cot(\theta) = \frac{7}{24}.

More problems from Find the constant of variation

Question
In a direct variation, y=15y=15 when x=5x=5. Write a direct variation equation that shows the relationship between xx and yy. Write your answer as an equation with yy first, followed by an equals sign.
Get tutor helpright-arrow

Posted 9 months ago

Question
In an inverse variation, y=4 y = 4 when x=6 x = 6 . Write an inverse variation equation that shows the relationship between x x and y y . Write the equation using a decimal or an integer. ____\_\_\_\_
Get tutor helpright-arrow

Posted 8 months ago

Question
The quantity z z varies directly with y y and inversely with x x . When y=−12 y = -12 and x=−4 x = -4 , z=−36 z = -36 . What is the equation of variation? Write your answer in the form A A or AB \frac{A}{B} , where A A and B B are constants or variable expressions. Remember to include the value of k k , the constant of variation, in exact form.\newline  z\ z =______
Get tutor helpright-arrow

Posted 9 months ago

Question
Which equation describes this relationship? Remember to include k k , the constant of variation. a a varies directly with b b and inversely with c c and d d \newlineChoices:\newline\[[A]a = \frac{kb}{cd}\]\newline\[[B]a = \frac{kbd}{c}\]\newline\[[C]a = \frac{k}{bcd}\]\newline\[[D]a = \frac{kd}{bc}\]
Get tutor helpright-arrow

Posted 9 months ago

Question
The quantity z z varies directly with y y and inversely with w w and x x . When y=12 y = 12 , w=8 w = 8 , and x=–5 x = –5 , z=–3 z = –3 . What is the value of k k , the constant of variation? \newlineWrite your answer in the form A A or (A)(B)\frac {(A)}{(B)} , where A A and B B are constants or variable expressions. \newline ____
Get tutor helpright-arrow

Posted 9 months ago

Question
Given the substitutions ln⁡2=a,ln⁡3=b \ln 2=a, \ln 3=b , and ln⁡5=c \ln 5=c , find the value of ln⁡(e3) \ln \left(\frac{e}{3}\right) in terms of a,b a, b , and c c .\newlineAnswer:
Get tutor helpright-arrow

Posted 9 months ago

Question
Square VWXY is dilated by a scale factor of 66 to form square V′W′X′Y′ V^{\prime} W^{\prime} X^{\prime} Y^{\prime} . Side W'X' measures 1212 . What is the measure of side WX?\newlineAnswer:
Get tutor helpright-arrow

Posted 9 months ago

Question
Solve for b b in the proportion.\newline108=b12b=□\begin{array}{l}\frac{10}{8}=\frac{b}{12} \\b=\square\end{array}
Get tutor helpright-arrow

Posted 6 months ago

Question
In a direct variation, y=4y = 4 when x=2x = 2. Write a direct variation equation that shows the relationship between xx and yy. \newline Write your answer as an equation with yy first, followed by an equals sign.
Get tutor helpright-arrow

Posted 6 months ago

Question
In a direct variation, y=18y = 18 when x=2x = 2. Write a direct variation equation that shows the relationship between xx and yy. \newline Write your answer as an equation with yy first, followed by an equals sign.
Get tutor helpright-arrow

Posted 6 months ago

View all (4)

Related topics

Algebra - Order of OperationsAlgebra - Distributive Property`X` and `Y` AxesGeometry - Scalene TriangleCommon MultipleGeometry - Quadrant