Front Matter1 Triangles and also Circles2 The Trigonometric Ratios3 legislations of Sines and also Cosines4 Trigonometric Functions5 Equations and Identities6 Radians7 one Functions8 an ext Functions and also Identities9 Vectors10 Polar collaborates and facility Numbers
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Section 4.1 Angles and also Rotation

Subsection Introduction

So far we have actually studied angles as parts of triangles, yet we can also use angles to describe rotation. For example, think that the minute hand on a clock. Every hour, the minute hand moves with one complete rotation, or (360degree ext.) In 2 hours, the minute hand rotates through (720degree ext.)

Example 4.1.

You are watching: How do you find the angle of rotation

Through how countless degrees does the minute hand turn in one hour and a half? In forty minutes?


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The volume control on an amplifier is a dial through ten settings, as shown at right. V how numerous degrees would you rotate the dial to rise the volume level indigenous 0 to 7?


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Subsection angles in standard Position

The degree measure that an angle depends just on the portion of a totality rotation between its sides, and also not on the location or position of the angle. To compare and also analyze angles, we ar them in typical position, so that the peak of the angle is located at the origin and also its initial side lies top top the optimistic (x)-axis. The figure below shows several angles placed in typical position.


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One-half a complete revolution is (180degree ext,) and three-quarters the one revolution is (270degree ext.) Thus, because that angles between (180degree) and (270degree) in conventional position, the terminal next lies in the third quadrant, and also for angles in between (270degree) and also (360degree ext,) the terminal next lies in the 4th quadrant.

Example 4.3.

Find the level measure of every angle presented below, and also sketch the edge in standard position.


Checkpoint 4.4.

Find the level measure of every angle below, and also sketch the edge in conventional position.


Subsection Trigonometric Ratios for all Angles

In thing 3 we identified the sine, cosine, and tangent for obtuse angle by placing the edge in a Cartesian coordinate system. We deserve to do the exact same for angles that represent rotations.


First, we place the angle ( heta) in traditional position, v its vertex at the origin. We picture the terminal side sweeping counter-clockwise roughly a circle to type the angle.

Next, we select a suggest (P) with collaborates ((x,y)) ~ above the terminal side, as displayed at right. The distance from the origin to (P) is then (r = sqrtx^2 + y^2 ext.) The trigonometric ratios the ( heta) are identified as follows.


The Trigonometric Ratios.

If ( heta) is an edge in typical position, and also ((x,y)) is a point on that is terminal side, v (r = sqrtx^2 + y^2 ext,) then


eginequation*lertsin heta = dfracyr~~~~~~~~~ cos heta = dfracxr~~~~~~~~~ an heta = dfracyxendequation*

We can choose any allude on the terminal next of the angle, and the trig ratios identified by its coordinates will be the same. (Can you describe why?)

Because it is the street from the beginning to (P ext,) (r) is always positive. However, (x) and (y) have the right to be confident or negative (or zero), relying on the angle ( heta ext.) for example, in the second quadrant, (x) is an unfavorable but (y) is positive, for this reason the cosine and also the tangent that angles in between (90degree) and also (180degree) room negative, but their sines space positive.

Example 4.5.

Give the sign of every of the 3 trigonometric ratios that the angles.

(displaystyle 200degree)

(displaystyle 300degree)


Solution.

a. In standard position, the terminal next of an angle of (200degree) lies in the 3rd quadrant. (See number (a) below.) In the third quadrant, (x lt 0) and also (y lt 0 ext,) yet (r gt 0 ext.) Thus, (sin 200degree) is negative, (cos 200degree) is negative, and also ( an 200degree) is positive.


b. The terminal next of (300degree) lies in the 4th quadrant, therefore (x gt 0) and (y lt 0 ext,) and also (r gt 0 ext.) Thus, (sin 300degree) is negative, (cos 300degree) is positive, and also ( an 300degree) is negative.


Checkpoint 4.6.

For angle in every of the four quadrants presented below, explain why the indicated trig ratios space positive. Then complete the table.


QuadrantDegreesSineCosineTangent
First(0degree lt heta lt 90degree)positivepositivepositive
Second(90degree lt heta lt 180degree)(hphantom0000)(hphantom0000)(hphantom0000)
Third(180degree lt heta lt 270degree)(hphantom0000)(hphantom0000)(hphantom0000)
Fourth(270degree lt heta lt 360degree)(hphantom0000)(hphantom0000)(hphantom0000)

Answer.
QuadrantDegreesSineCosineTangent
First(0degree lt heta lt 90degree)positivepositivepositive
Second(90degree lt heta lt 180degree)positivenegativenegative
Third(180degree lt heta lt 270degree)negativenegativepositive
Fourth(270degree lt heta lt 360degree)negativepositivenegative

Example 4.7.

Find the sine, cosine, and also tangent that the angle displayed at right.


Solution.

The (y)-coordinate of the allude (P) is (-5 ext,) and (r = 6 ext,) so


eginequation*sin heta = dfracyr = dfrac-56endequation*

To uncover the (x)-coordinate of (P ext,) we use the equation the a one of radius (6 ext,) (x^2 + y^2 = 36 ext.)


eginalign*x^2 + (-5)^2 amp = 36\x^2 amp = 36 - 25 = 11\x amp = pm sqrt11endalign*

Because (P) is in the third quadrant, (x = -sqrt11 ext.) Thus,


eginequation*cos heta = dfracxr = dfrac-sqrt116~~~~ extand ~~~~ an heta = dfracyx = dfrac-5-sqrt11 = dfrac5sqrt11endequation*
Checkpoint 4.8.

Find the sine, cosine, and also tangent of the angle displayed at right. The circle has radius 4.


Answer.

(sin heta = dfracsqrt74,~~cos heta = dfrac-34,~~ an heta = dfrac-sqrt73)


Subsection referral Angles

In section 3.1 we learned the the trig ratios for angles in the second quadrant space the exact same as the trig ratios for your supplements, other than for sign. Because that example, you can use her calculator to verify that


(sin 130degree = 0.7660)(sin 50degree = 0.7660)
(cos 130degree = -0.6428)(cos 50degree = 0.6428)
( an 130degree = -1.1918)( an 50degree = 1.1918)

The trig ratios because that (130degree) and (50degree) have actually the same absolute value since the 2 triangles formed by the angles space congruent, as displayed above.

( riangle OP^primeQ^prime) is referred to as a recommendation triangle because that (130degree ext,) and also (50degree) is dubbed the recommendation angle.

The trig ratios for angles in between (180degree) and also (360degree ext,) whose terminal sides lie in the 3rd and 4th quadrants, are additionally related come the trig ratios of familiar angles in the first quadrant. We "refer" the angle to a first quadrant angle v a congruent reference triangle.

Note 4.9.

Reference angles will be crucial when we must solve trigonometric equations. There will constantly be two angles through the same trig ratio (except for quadrantal angles), and they will have the same reference angle.

We can construct referral triangles for angle in any of the four quadrants, and also the trig ratios the the angle space the exact same as the trig ratios of its reference angle, up to sign. Right here is just how to build a recommendation triangle because that an edge :

Choose a allude (P) ~ above the terminal side.

Draw a line from point (P) perpendicular come the (x)-axis.

The figure below shows angles ( heta) in all four quadrants, and the reference angle, (widetilde heta ext,) because that each. Examine the figures, and also make sure you know the formula for finding the referral angle in each quadrant.


Use a protractor to draw an edge of (56degree) in traditional position. Draw its reference triangle.

Use your calculator to uncover the sine and cosine the (56degree ext,) rounded to two decimal places. Brand the sides of the reference triangle v their lengths.

What room the works with of the allude (P) whereby your angle intersects the unit circle?

Draw the enjoy of your recommendation triangle throughout the (y)-axis, so the you have actually a congruent triangle in the 2nd quadrant.

You now have actually the referral triangle because that a second-quadrant angle in traditional position. What is the angle?

Use her calculator to find the sine and also cosine of your brand-new angle. Brand the collaborates of the point (Q) wherein the angle intersects the unit circle.

Draw the enjoy of her triangle from component (4) across the (x)-axis, so the you have actually a congruent triangle in the 3rd quadrant.

You now have actually the referral triangle for a third-quadrant angle in standard position. What is that angle?

Use her calculator to discover the sine and cosine of your new angle. Label the works with of the point (R) whereby the edge intersects the unit circle.

Draw the have fun of your triangle from part (7) throughout the (y)-axis, so the you have a congruent triangle in the 4th quadrant.

You now have the recommendation triangle for a fourth-quadrant angle in traditional position. What is that angle?

Use your calculator to find the sine and also cosine of your new angle. Brand the collaborates of the allude where the angle intersects the unit circle.

Generalize: All 4 of her angles have actually the same referral angle, (56degree ext.) for each quadrant, create a formula because that the edge whose referral angle is ( heta ext.)

Quadrant I:

Quadrant II:

Quadrant III:

Quadrant IV:

Example 4.11.

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Find the referral angle because that (200degree ext.)

Sketch (200degree) and also its reference angle in typical position, along with their recommendation triangles. Verify that both angles have actually the very same trigonometric ratios, as much as sign.