Trig Angle Approximations

Question

Angles α and β are between 0° and 360° with cos α = -0.9063 and tan β = -0.4877.
Which of the following is a possible (approximate) value for α + β?

A) 51° B) 129° C) 309° D) 361° E) 669°

Answer 1

Angles α and β are between 0° and 360° with cos α = -0.9063 and tan β = -0.4877.
Which of the following is a possible (approximate) value for α + β?

 

 

Since cos α is negative, then α must either be a quadrant II or quadrant III angle.

Since tan β is negative, then β must either be a quadrant II angle or a quadrant IV angle. 

 

 

α + β is therefore a sum with one of the following combinations

1. Quadrant II angle + Quadrant II angle   180 < α + β < 360

2. Quadrant II angle + Quadrant IV angle  360 < α + β < 540

3. Quadrant III angle + Quadrant II angle  270 < α + β < 450

4. Quadrant III angle + Quadrant IV angle 450 < α + β < 630

 

Finding the values of  α and β using a calculator, then  the choices given, only 309^o satisfies one of the possibilities described above.

Answer 2

First we need to find α and β.  Approximate cosα so that the angle is on a quadrant.

 

α = cos^-1(-.9063)                   β = tan^-1(-1/2)

 

α = 155                                 β = 25

 

 

cosine and tangent are both negative in a common quadrant.  That is quadrant 2.

 

α = 155                                β = 180 - 25

α = 180 - 155                       β = 155

α = 25

 

 

α + β = 25 + 155 = 180