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.

   

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]

   

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]

    (Video) Trigonometry For Beginners!

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

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

[latex]\displaystyle 600°[/latex]

[latex]\displaystyle -200°[/latex]

[latex]\displaystyle -700°[/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.

[latex]\displaystyle -25°[/latex]

[latex]\displaystyle 100°[/latex]

[latex]\displaystyle 250°[/latex]

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

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

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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.

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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.

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

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

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For tasks 51 to 54,

1. Draw the function.

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

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

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