find all possible values for t in the interval 0=t=2p . If there is more than one answer, enter your solutions in a comma separated list.

Question

find all possible values for t in the interval 0≤t≤2pi If there is more than one answer, enter your solutions as a comma seperated list.

1)sin(t)=1/2 when t=            (I know one answer is pi/6, but it tells me there is another answer too

2) cos(t)=-1/2 when t=         (I know one answer is 2pi/3, but ther is another answer too)

3) tan(t)=1/sqrt3 when t=     (I know one answer is pi/6, but ther is another answer too)

Answer 1

1) sine is positive in the first and second quadrant

    sin pi/6=30º

    30º is the reference angle

    form a 30º angle with the x-axis in quadrant  two

    180º-30º=150º or 5pi/6

 

2) cosine is negative in the second and third quadrants

    cos 2pi/3=120º

    make an angle of 120º in the third quadrant

    this angle is made with the positive x-axis and a segment in the third quadrant

    360º-120º=240º or 4pi/3

 

3) tangent=sine/cosine

    tangent is positive in the first and third quadrants because sin and cos are both positive in the first quadrant and sin and cos are both negative in the third quadrant

   tan pi/6=30º

   make a 30º angle in the third quadrant with the negative x-axis

   180º+30º=210º or 7pi/6

    tan210=sin210/cos210=(-1/2)/(-√3/2)=1/√3

 

 

Answer 2

5pi / 6 for a

Answer 3

I don't know how much this will help you because it's not that easy to explain in writing. Did your teacher tell you about All Students Take Calculus or something to that effect? This stands for all trig functions are positive in quadrant 1, Sine is positive in quadrant 2, tangent is positive in quadrant 3 and cosine is positive in quadrant 4. So for your first problem, yes, pi/6 is the answer in the quadrant 1. It is also your reference angle for the other quadrants. Since sine comes out as positive 1/2, there is also an answer in quadrant 2. A 30 degree reference angle in quadrant 2 is 150 degrees or 5pi/6. By the way, a reference angle is always formed by the x axis. So for problem 3, pi/6  is the reference angle again but this time tangent isalso positive in quadrant 3. so a 30 degree reference angle in quadrant 3 is 210 degrees or 7pi/6. Now problem 2 is a little different because the angle you got from the calculator is not the reference angle. 2pi/3 is 120 degrees so the reference angle is 60. Since cosine is negative, you need to check quadrant 2 and 3. (we're still using A-S-T-C) So a 60 degree reference angle in quadrant 3 is 240 or 4pi/3. The calculator always give you the reference angle when the original expression was positive but it DOES NOT give you the reference angle when the original expression was negative. You need to figure out the reference angle and then check the 2 appropriate quadrants.