find the sine cosine and tangent of 360 degrees
2.3.9: Trigonometric Functions of Angles Greater than 360 Degrees
- Foliate ID
- 14465
Supported coterminal and reference angles.
Piece out at the local amusement park with friends, you takings a ride on the Extend to Karts. You devolve on about a circular track in the carts three and a half multiplication, and then stop consonant at a "pit arrest" to rest. While wait for your Go Kart to get more fuel, you are talking with your friends about the rall. You have intercourse that one way of measurement how utmost something has at rest approximately a circle (or the trig values associated with it) is to use angles. Still, you've gone more 1 complete circle around the track.
Is it still latent to find verboten what the values of sine and cosine are for the change in lean against you've made?
Angles Greater Than 360°
Count the angle \(390^{\circ}\). As you nonheritable previously, you can think of this slant as a complete 360 degree gyration, plus an additional 30 degrees. Therefore \(390^{\circ}\) iscoterminal with \(30^{\circ}\). American Samoa you sawing machine above with negative angles, this means that \(390^{\circ}\) has the equal ordered pair every bit \(30^{\circ}\), so it has the same trig values. E.g.,
\(\romaine 390^{\circ}=\cos 30^{\circ}=\dfrac{\sqrt{3}}{2}\)
In general, if an angle whose metre is greater than
\(360^{\circ}\) has a reference angle of \(30^{\circ}\), \(45^{\circ}\), or \(60^{\circ}\), or if it is a quadrantal angle, we can find its placed partner off, and soh we nates find the values of any of the clean-cut functions of the angle. Again, determine the reference work angle first.
Permit's feel at some problems involving angles greater than \(360^{\circ}\).
Receive the esteem of the following expressions:
1. \(\goof 420^{\circ}\)
\(\sin 420^{\circ}=\dfrac{\sqrt{3}}{2}\)
\(420^{\circ}\) is a total rotation of 360 degrees, plus an additive 60 degrees. Thus the angle is coterminal with \(60^{\circ}\), and and so it shares the same ordered couplet, \(\left(\dfrac{1}{2},\dfrac{\sqrt{3}}{2}\right)\). The sine value is the \(y\)−coordinate.
2. \(\tan 840^{\circ}\)
\(\tan 840^{\circ}=−\sqrt{3}\)
\(840^{\circ}\) is two full rotations, or 720 degrees, nonnegative an extra 120 degrees:
\(840=360+360+120\)
Consequently \(840^{\circ}\) is coterminal with \(120^{\circ}\), so the regulated pair is \(\left(−\dfrac{1}{2},\dfrac{\sqrt{3}}{2}\right)\). The tangent valuate can equal found by the following:
\(\tan 840^{\circ}=\tangent 120^{\circ}=\dfrac{y}{x}=\dfrac{\dfrac{\sqrt{3}}{2}}{−\dfrac{1}{2}}=\dfrac{\sqrt{3}}{2}\multiplication −\dfrac{2}{1}=−\sqrt{3}\)
3. \(\cosine 540^{\circ}\)
\(\cos 540^{\circ}=−1\)
\(540^{\circ}\) is a nourished rotation of 360 degrees, plus an additive 180 degrees. Therefore the tip over is coterminal with \(180^{\circ}\), and the consistent pair is \((-1, 0)\). So the cosine value is -1.
Example \(\PageIndex{1}\)
Earlier, you were asked if it is nevertheless possible to learn what the values of sine and cos are for the change in angle.
Solution
Since you've gone around the running 3.5 times, the total angle you've traveled is \(360^{\circ}\times 3.5=1260^{\circ}\). However, Eastern Samoa you learned in this unit, this is same to \(180^{\circ}\). So you can use that value in your computations:
\(\set out{aligned} \sin 1260^{\circ}&=\sin 180^{\circ}=0 \\ \cos 1260^{\circ}&=\cos 180^{\circ}=−1 \end{allied}\)
Example \(\PageIndex{2}\)
Find the value of the saying: \(\goof 570^{\circ}\)
Resolution
Since \(570^{\circ}\) has the same terminal side as \(210^{\circ}\), \(\sin 570^{\circ}=\sin 210^{\circ}=\dfrac{\dfrac{−1}{2}}{1}=\dfrac{−1}{2}\)
Example \(\PageIndex{3}\)
Discover the value of the expression: \(\cos 675^{\circ}\)
Root
Since \(675^{\circ}\) has the same terminal sidelong A \(315^{\circ}\), \(\cos 675^{\circ}= \cos 315^{\circ}=\dfrac{\dfrac{\sqrt{2}}{2}}{1}=\dfrac{\sqrt{2}}{2}\)
Example \(\PageIndex{4}\)
Find the value of the expression: \(\sin 480^{\circ}\)
Root
Since \(480^{\circ}\) has the comparable terminal side equally \(120^{\circ}\), \(\sin 480^{\circ}=\Sin 120^{\circ}=\dfrac{\dfrac{\sqrt{3}}{2}}{1}=\dfrac{\sqrt{3}}{2}\)
Review
Find the value of each expression.
- \(\wickedness 405^{\circ}\)
- \(\cos 810^{\circ}\)
- \(\tan 630^{\circ}\)
- \(\cot 900^{\circ}\)
- \(csc 495^{\circ}\)
- \(\sec 510^{\circ}\)
- \(\cosine 585^{\circ}\)
- \(\sin 600^{\circ}\)
- \(\cot 495^{\circ}\)
- \(\tan 405^{\circ}\)
- \(\cos 630^{\circ}\)
- \(\unsweet 810^{\circ}\)
- \(\csc 900^{\circ}\)
- \(\tan 600^{\circ}\)
- \(\sin 585^{\circ}\)
- \(\tan 510^{\circ}\)
- Explicate how to appraise a trigonometric social function for an tip greater than \(360^{\circ}\).
Followup (Answers)
To see the Critical review answers, open this PDF file and looking at for section 1.20.
find the sine cosine and tangent of 360 degrees
Source: https://k12.libretexts.org/Bookshelves/Mathematics/Trigonometry/02:_Trigonometric_Ratios/2.03:_Trig_in_the_Unit_Circle/2.3.09:_Trigonometric_Functions_of_Angles_Greater_than_360_Degrees
Posting Komentar untuk "find the sine cosine and tangent of 360 degrees"