Friday, October 7, 2011

solving for values of sine, cosine and tangent.

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.


2 comments:

  1. I think you made a mistake on answer 3. 360 - 19.471 = 340.529

    I got .1082pi and 1.8918pi.

    On number 4 when you factor sinx - sinx/cosx, shouldn't it be sinx(1 - 1/cosx) = 0 ?

    ReplyDelete
  2. Amending the corrections from Nemo. Thanks.


    1) 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°

    ReplyDelete