Let's assume we can solve a trigonometric equation to get a value for sina, cosa or tana. We need only use the corresponding inverse function to get the angle a.
For any of these simple type of problems, we are only half way home if we are required to find all the angles between 0° and 360°, or if we are measuring in radians, between 0 and 2pi.
Let's assume for argument's sake that the angle you get is a in our picture, something in first quadrant. (It's possible if cosa is negative, the first angle we will get is in the second quadrant. If cosa < 0 or tana < 0, the angle will be negative degrees or radians and in the fourth quadrant.
The case for cosine: If cosa = constant, then cos(360-a)° = constant. (If you are dealing with radians, it's a and 2pi - a.)
The case for sine: If sina = constant, then sin(180-a)° = constant. (If you are dealing with radians, it's a and pi - a.)
The case for tangent: If tana =
constant, then tan(180+a)° = constant. (If you are dealing with
radians, it's a and pi + a.)
Problems. Give the answers as decimal degrees, rounded to the nearest thousandth of a degree and radians as k(pi), where k is rounded to four places after the decimal.
1) sina = .9
2) tanb = -2.5
3) cosc = sqrt(8)/3
*4) sinx = tanx
Answers in the comments.
The case for sine: If sina = constant, then sin(180-a)° = constant. (If you are dealing with radians, it's a and pi - a.)
Problems. Give the answers as decimal degrees, rounded to the nearest thousandth of a degree and radians as k(pi), where k is rounded to four places after the decimal.
1) sina = .9
2) tanb = -2.5
3) cosc = sqrt(8)/3
*4) sinx = tanx
Answers in the comments.
I think you made a mistake on answer 3. 360 - 19.471 = 340.529
ReplyDeleteI got .1082pi and 1.8918pi.
On number 4 when you factor sinx - sinx/cosx, shouldn't it be sinx(1 - 1/cosx) = 0 ?
Amending the corrections from Nemo. Thanks.
ReplyDelete1) sina = .9
sin inverse of .9 rounds to 64.158°. Subtracting this from 180° gives us 115.842°
Divide the angles by 180° to get the number that multiplies pi.
.3564pi and .6436pi
2) tanb = -2.5
tan inverse of -2.5 rounds to -68.199°. If we want answers between 0° and 360°, we add 180° to get 111.801° and add 360° to get 291.801°
Divide the angles by 180° to get the number that multiplies pi.
.6211pi and 1.6211pi
3) cosc = sqrt(8)/3
cos inverse of sqrt(8)/3 rounds to 19.471°. Subtracting from 360° gives us 340.529°
Divide the angles by 180° to get the number that multiplies pi.
0.1091pi and 1.8919pi
*4) sinx = tanx
This one takes more work. Let's subtract to set one side equal to zero.
sinx - tanx = 0
since tanx = sinx/cosx, we van re-write as
sinx - sinx/cosx = sinx(1-1/cosx) = 0
Two things multiply to 0 means one or the other is zero.
sinx = 0 at 0° and 180°.
or 0 and pi when measured in radians.
1-1/cosx = 0 means cosx = 1 which is at 0° as well, so there are the two answers at 0° and 180°