Trigonometry Summary and Review Chapter 4: Trigonometry (2023)

Section 4.4 Chapter 4 Summary and Review

key concepts

  1. We can use angles to describe rotation. Positive angles indicate counterclockwise rotation; negative angles describe a clockwise rotation.

  2. We define the trigonometric ratios of any angle by putting the angle in the standard position and picking a point on the connection side, with [latex]r = \sqrt{x^2 + y^2}{.}[/latex]

    trigonometric ratios.

    If [latex]\theta[/latex] is an angle in standard position and [latex](x,y)[/latex] is a point on its end face, then [latex]r = \sqrt{x^2 + y ^2}{,}[/latex] then

    [latex]\sin \theta=\dfrac{y}{r}~~~~~~~~\cos \theta=\dfrac{x}{r}~~~~~~~~\tan\ theta = \dfrac{y}{x}[/latex]

  3. To construct a datum triangle for an angle:

    1. Choose a point [latex]P[/latex] on the terminal page.

    2. Draws a line from the point [latex]P[/latex] perpendicular to the [latex]x[/latex] axis.

  4. Isreference anglefor [latex]\theta[/latex] is the positive acute angle formed between the end face of [latex]\theta[/latex] and the [latex]x[/latex] axis.

    Trigonometry Summary and Review Chapter 4: Trigonometry (1)

  5. The trigonometric ratios of each angle are equal to the ratios of its reference angle, except for the sign. The sign of the ratio is determined by the quadrant.

  6. Find an angle [latex]\theta[/latex] with a given reference angle [latex]\widetilde{\theta}{:}[/latex].

    Cuadrante I: [latex]~~~~~~\theta = \widetilde{\theta}[/latex][latex]\hghost{0000}[/latex]Cuadrante II: [latex]~~~~\theta = 180° - \widetilde{\theta}[/latex]
    Quadrant III: [latex]~~~~~\theta = 180° + \widetilde{\theta}[/latex][latex]\hghost{0000}[/latex]Cuadrante IV: [latex]~~~~\theta = 360° - \widetilde{\theta}[/latex]

    Trigonometry Summary and Review Chapter 4: Trigonometry (2)

  7. There are always two angles between [latex]0°[/latex] and [latex]360°[/latex] (except quadrant angles) with a given trigonometric ratio.

  8. coterminal StoreThey have the same trigonometric ratios.

  9. Solve an equation of the form [latex]\sin \theta = k{,}[/latex] or [latex]\cos \theta = k{,}[/latex] or [latex]\tan \theta = k { ,}[/latex] we can use the corresponding inverse trigger key on a calculator to find a solution (or coterminal angle). We use reference angles to find a second solution between [latex]0°[/latex] and [ latex]360°{.}[/latex].

  10. Angles in a unit circle.

    Let [latex]P[/latex] be a point on a unit circle determined by the end face of an angle [latex]\theta[/latex] in standard position. Then the coordinates [latex](x,y)[/latex] of [latex]P[/latex] are given by

    [latex]x = \cos \theta,~~~~~~y = \sin \theta[/latex]

  11. coordinates.

    If the point [latex]P[/latex] is a distance [latex]r[/latex] from the origin in the direction given by the angle [latex]\theta[/latex] at the default position, then the coordinates of [ latex ]P[/latex] are

    [latex] x = r \cos \theta ~~~~ {y} ~~~~ y = r \sin \theta[/latex]

  12. Navigational instructions for ships and airplanes are sometimes given asAspects, which are angles measured clockwise from north.

  13. Periodic functions are used to model phenomena that exhibit cyclic behavior.

  14. The trigonometric ratios [latex]\sin \theta[/latex] and [latex]\cos \theta[/latex] are functions of the angle [latex]\theta{.}[/latex]

  15. IsPeriodof the sine function is [latex]360°{.}[/latex] Itscenterlineis the horizontal line [latex]y = 0{,}[/latex] and theAmplitudethe sine function is 1.

  16. The graph of the cosine function has the same period, center line, and amplitude as the graph of the sine function. However, the locations of the intersections and the maximum and minimum values ​​are different.

  17. We use the notation [latex]y = f(x)[/latex] to indicate that [latex]y[/latex] is a function of [latex]x{,}[/latex], so [latex] x [ /latex] is the input variable and [latex]y[/latex] is the output variable.

  18. The tangent function has the period [latex]180°{.}[/latex] It is not defined on odd multiples of [latex]90°{,}[/latex] and increases with each interval of its value range.

  19. tilt angle.

    Istilt angleof a line is the angle [latex]\alpha[/latex] measured in the positive direction from the positive [latex]x[/latex] axis to the line. If the slope of the line is then [latex]m{,}[/latex].

    [latex]\tan \alpha = m[/latex]

    otherwise [latex]0° \alpha \alpha 180°{.}[/latex]

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  20. amplitude, period and center line.

    1. The graph of

      [latex]y = A\cos\theta ~~{o}~~ y = A\sin\theta[/latex]

      hat Amplitude [latex]\abs{A}{.}[/latex]

    2. The graph of

      [latex] y = \cos B\theta ~~{o}~~ y = \sin B\theta[/latex]

      tiene punto [latex]\dfrac{360°}{\abs{B}}{.}[/latex]

    3. The graph of

      [latex]y = k + \cos\theta ~~{o}~~ y =k + \sin\theta[/latex]

      has center line [latex]y = k{.}[/latex]

  21. The graph of [latex]y = k + A\sin B\theta[/latex] has amplitude [latex]A{,}[/latex] period [latex]\dfrac{360°}{B}{,}[ /latex] and center line [latex]y = k{.}[/latex] The same applies to the graph of [latex]y = k + A\cos B\theta{.}[/latex]

  22. Functions that have graphs in the form of sine or cosine are calledsinusoidal.

  23. periodic function.

    The function [latex]y = f(x)[/latex] isZeitungif there is a smaller value of [latex]p[/latex] such that

    [latex]f(x + p) = f(x)[/latex]

    for all [latex]x{.}[/latex] The constant [latex]p[/latex] is calledPeriodthe function

Exercises Chapter 4 Review exercises


The London Eye, the world's largest Ferris wheel, rotates once every 30 minutes. How many degrees will it rotate in 1 minute?


The London Eye in Problem 1 has 32 cabins evenly spaced along the wheel. If the cabins are numbered 1 through 32, what is the angular distance between cabin number 1 and number 15?

exercise group.

For tasks 3 and 4, find two angles, one positive and one negative, that are coterminal with the given angle.

  1. [latex]\displaystyle 510°[/latex]

[latex]\displaystyle 600°[/latex]

[latex]\displaystyle -200°[/latex]

[latex]\displaystyle -700°[/latex]

  1. [latex]\displaystyle -380°[/latex]

  2. [latex]\displaystyle -423°[/latex]

  3. [latex]\displaystyle 187°[/latex]

  4. [latex]\displaystyle 1000°[/latex]

exercise group.

Give the corresponding quadrant and the reference angle for the angles in exercises 5 and 6. Give three more angles using the same reference angle, one for each of the other three quadrants. Draw the four angles.

  1. [latex]\displaystyle -300°[/latex]

[latex]\displaystyle -25°[/latex]

[latex]\displaystyle 100°[/latex]

[latex]\displaystyle 250°[/latex]

  1. [latex]\displaystyle 430°[/latex]

  2. [latex]\displaystyle -590°[/latex]

  3. [latex]\displaystyle -95°[/latex]

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  4. [latex]\displaystyle 1050°[/latex]


Let [latex]\widetilde{\theta} = f(\theta)[/latex] be the function that gives the reference angle of [latex]\theta{.}[/latex]. Example: [latex]f(110°) = 70°[/latex] because the reference angle for [latex]110°[/latex] is [latex]70°{.}[/latex].

  1. Complete the table of values.

  2. Choose appropriate scales for the axes and plot the function for [latex]-360° \le \theta \le 360°{.}[/latex]

    Trigonometry Summary and Review Chapter 4: Trigonometry (3)


Let [latex]\widetilde{\theta} = f(\theta)[/latex] be the function that gives the reference angle of [latex]\theta{.}[/latex] (see Exercise 7). Is [latex ]f[/latex] a periodic function? If yes, provide period, centerline and amplitude. If not, explain why not.

exercise group.

For exercises 9 to 20, solve the equation for [latex]0° \le \theta \le 360°{.}[/latex]


[latex]\sin \theta = \dfrac{-1}{2}[/latex]


[latex]\cos \theta = \dfrac{-1}{\sqrt{2}}[/latex]


[latex]2\cos\theta + 1 = 0[/latex]


[latex]5\sin θ + 5 = 0[/latex]


[latex]\tan\theta - 1 = 0[/latex]


[latex]\sqrt{3} + 3\tan \theta = 0[/latex]


[latex]\cos\theta = \cos(-23°)[/latex]


[Látex]\sin \theta = \sin(370°)[/latex]


[latex]\tan \theta = \tan 432°[/latex]


[latex]\tan \theta = \tan (-6°)[/latex]


[Látex]\sin \theta + \sin 83° = 0[/latex]


[latex]\cos \theta + \cos 429° = 0[/latex]

exercise group.

For exercises 21 to 26, solve the equation for [latex]0° \le \theta \le 360°{.}[/latex] Round your answers to two decimal places.


[latex]3\sin θ + 2 = 0[/latex]


[latex]5\cos\theta + 4 = 0[/latex]




[latex]-4\tan\theta + 12 = 0[/latex]


[latex]4 = 8\cos θ + 9[/latex]


[latex]-3 = 6\sin \theta -5[/latex]

exercise group.

For tasks 27 to 32, find the coordinates of the point where the end face of the angle [latex]\theta[/latex] in the standard position intersects the circle of radius [latex]r[/latex] pointing to the origin is centered. Round your answers to two decimal places.


[latex]\theta = 193°,~ r = 10[/latex]

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[latex]\theta = -12°,~ r = 20[/latex]


[latex]\theta = 92°,~ r = 8[/latex]


[latex]\theta = 403°,~ r = 6[/latex]


[latex]\theta = -341°,~ r = 3[/latex]


[latex]\theta = -107°,~ r = 20[/latex]


If you follow a course of [latex]190°[/latex] for 10 miles, how far south and west are you from your starting point? Round to two decimal places.


A boy releases a balloon, which then follows a course of [latex]84°[/latex] for 150 meters. How far east and how far north is the balloon from where it was launched? Round to the nearest meter.

exercise group.

For problems 35-38, write the equation of a sine or cosine function with the given properties and sketch the graph, including at least one period.


amplitude 7, center line [latex]y = 4{,}[/latex] period 2, [latex]y[/latex] section 4


amplitude 100, center line [latex]y = 50{,}[/latex] period 12, [latex]y[/latex] section 100


Maximum point at [latex](90°, 24)[/latex] and minimum point at [latex](270°, 10)[/latex]


Horizontal intersection at [latex]180°{,}[/latex] Maximum points at [latex](0°, 5)[/latex] and [latex](360°, 5)[/latex]

exercise group.

For exercises 39-42, evaluate the expression for [latex]f(\theta) = \sin \theta[/latex] and [latex]g(\theta) = \cos \theta{.}[/latex]. .


[latex]2f(\theta)g(\theta)[/latex] für [latex]\theta = 30°[/latex]


[latex]f(\dfrac{\theta}{2})[/latex] für [latex]\theta = 120°[/latex]


[latex]g(4\theta) - f(2\theta)[/latex] für [latex]\theta = 15°[/latex]


[Latex]\dfrac{1 - g(2\theta)}{2}[/latex] für [Latex]\theta = 45°[/latex]

exercise group.

For exercises 43-46, write an equation for the given graph and give the exact coordinates of the marked points.


Trigonometry Summary and Review Chapter 4: Trigonometry (4)


Trigonometry Summary and Review Chapter 4: Trigonometry (5)


Trigonometry Summary and Review Chapter 4: Trigonometry (6)


Trigonometry Summary and Review Chapter 4: Trigonometry (7)


Every 24 hours, Delbert takes 50 mg of a therapeutic drug. The level of this drug in Delbert's bloodstream immediately jumps to its peak of 60 mg, but the level falls to its nadir of 10 mg just before the next dose.

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  1. Draw a graph of [latex]d(t){,}[/latex] the amount of drug in Delbert's bloodstream after one of his doses. Show at least three cans.

  2. What is the period of [latex]d(t){?}[/latex]


A water fountain has water dripping into a bowl, but once the bowl is full, it tips over and the water spills out. The container is then tipped back and begins to fill again. The container is filled 5 times per minute.

  1. Draw a diagram of [latex]h(t){,}[/latex] the current height of the water in the tank. Display at least two full fill and drain cycles.

  2. What is the period of [latex]h(t){?}[/latex]


Henry watches Billie ride the carousel. It stands 2 meters from the carousel, which is 10 meters in diameter. He notices that she walks past him three times every minute.

  1. Draw a graph of [latex]f(t){,}[/latex] the distance between Henry and Billie at time [latex]t{.}[/latex] Show at least two complete circuits.

  2. What is the period of [latex]f(t){?}[/latex]


An ant runs along the triangle with vertices at [latex](0,0),~(1,1),[/latex] and [latex](0, 1) { .} [/Latex]

  1. How far does the ant have to walk to get from [latex](0,0)[/latex] to [latex](1,1){?}[/latex] From [latex](1,1)[/ latex ] to [latex](0,1){?}[/latex] From [latex](0,1)[/latex] to [latex](0,0){?}[/latex]

  2. Draw a graph of [latex]g(t){,}[/latex] the [latex]y[/latex] coordinate of the ant at time [latex]t{.}[/latex] Show at least two circuits the triangle.

  3. What is the period of [latex]g(t){?}[/latex]

exercise group.

For tasks 51 to 54,

  1. Draw the function.

  2. Specify the amplitude, period, and center line of the function.


[latex]y = 4 + 2 \cos \theta[/latex]


[latex]y = -1 + 3 \cos \theta[/latex]


[Latex] y = 1,5 + 3,5 \sen 2\theta[/latex]


[latex]y = 1,6 + 1,4 \sin (0,5\theta)[/latex]

exercise group.

For questions 55-58 find the angle of inclination of the line.


[latex] y = \dfrac{\sqrt{3}}{3} x + 1[/latex]


[latex]y = - x - 11[/latex]


[latex]y = 100 - 28 x[/latex]


[latex]y = - 3,7 + 1,4x[/latex]

exercise group.

For exercises 59-62 find an equation for the straight line through the given point with angle of inclination [latex]\alpha{.}[/latex]


[latex](0,2){,}[/latex] [latex]~ \alpha = 45°[/latex]


[latex](4,0){,}[/latex] [latex]~ \alpha = 135°[/latex]


[latex](3,-4){,}[/latex] [latex]~ \alpha = 120°[/latex]


[latex](-7,2){,}[/latex] [latex]~ \alpha = 60°[/latex]


Draw the graphs of [latex]y = \tan \theta[/latex] and [latex]y = \cos \theta[/latex] on the same grid for [latex]-180° \lt \theta \lt 180° {.}[/latex] How are the [latex]\theta[/latex] intercepts of the graph of [latex]y = \cos \theta[/latex] related to the graph of [latex]y = \tan \ theta[/latex] ?


Draw the graphs of [latex]y = \tan \theta[/latex] and [latex]y = \sin \theta[/latex] on the same grid for [latex]-180° \lt \theta \lt 180° {.}[/latex] How are the [latex]\theta[/latex] intercepts of the graph of [latex]y = \sin \theta[/latex] related to the graph of [latex]y = \tan \ theta[/latex] ?


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