Simple Trigonometric Equations

Trigonometric Equations

1. Solve:    sin x = .5   for 0^o< x < 360^o

                       x = arcsin .5
                   x = 30^o, 150^o
Use your special angle values to determine x.  Then consider other angles in the domain that might have the same sine value.
2. Solve:   cos x = .7134    for    0^o < x < 360^o

                        x = arccos .7134
                  x = 44.488^o, 315.512^o
Use your calculator to calculate cos^-1.7134, using degree mode.  Then consider what quadrant the second answer should come from. Since the cosine value is positive, it must come from quadrant IV. Using the reference angle 44.488, calculate 360 - 44.488.  Actually, type 360 - ANS.
3. Solve:  sec x = 1.3452   for   0 < x < 2p

           cos x = 1/1.3452
                 x = arccos (1/1.3452)
                 x = .7327 radians, 5.5505 radians
Put your calculator in radian mode and calculate the first answer.  The secant is also positive in the fourth quadrant.  Using .7327 as the reference angle, the fourth quadrant angle is 2p - .7327 or 2p - ANS.

Angle of Inclination and Slope

Theorem:
For any line with slope m and inclination a,
                       m = tan a     if a is not 90^o
If a = 90^o, then the line has no slope. (The line is vertical.)

Example:
To the nearest degree, find the inclination of the line 2x + 5y = 15.

y = (-2/5) x + 3 (rewriting the equation)
m = -2/5 = tan a
a = Tan^-1a = 21.8^o
Since tan a is negative and 0^o< a < 180^o,then a must be 180^o - 21.8^o = 158.2^o.

Problems