if sin s = -5 /13
and sin t -3/5
s is in QIII
t is in QIV
find tan (s-t)
i got that tan (s-t)= 56/53
is that right? can someone check my work?
If sin s = -5/13, and it is in the third quadrant, cos s = -12/13
s = 202.620 degrees
If sin t is -3/5, and t is in the fourth quadrant, cos t = 4/5.
t = 323.130 degrees
tan (s-t) = sin (s-t)/cos(s-t)
= [sin s cos t - sin t cos s]/
[cos s cos t + sin s sin t]
Crunch the numbers. You should get a rational fraction, but your 53 in the denominator looks fishy.
You answer should be 56/33.
To find the value of tan (s - t), you need to use the trigonometric identity:
tan (s - t) = (tan s - tan t) / (1 + tan s * tan t)
First, let's find the values of tan s and tan t:
Given that sin s = -5/13 and s is in Quadrant III, we can determine cos s using the Pythagorean identity: sin^2 s + cos^2 s = 1.
cos^2 s = 1 - sin^2 s
cos^2 s = 1 - (-5/13)^2
cos^2 s = 1 - 25/169
cos^2 s = 144/169
Since s is in Quadrant III, cos s is negative. Therefore:
cos s = -12/13
Now, we can find tan s using the ratio: tan s = sin s / cos s
tan s = (-5/13) / (-12/13)
tan s = 5/12
Similarly, we can find tan t using the given information: sin t = -3/5, t is in Quadrant IV.
cos t = √(1 - sin^2 t)
cos t = √(1 - (-3/5)^2)
cos t = √(1 - 9/25)
cos t = √(16/25)
cos t = 4/5
Since t is in Quadrant IV, both sin t and cos t are positive. Therefore:
tan t = sin t / cos t
tan t = (-3/5) / (4/5)
tan t = -3/4
Now, substitute these values into the trigonometric identity:
tan (s - t) = (tan s - tan t) / (1 + tan s * tan t)
tan (s - t) = (5/12 - (-3/4)) / (1 + (5/12)(-3/4))
tan (s - t) = (5/12 + 3/4) / (1 - 15/48)
tan (s - t) = (5/12 + 9/12) / (1 - 15/48)
tan (s - t) = 14/12 / (1 - 15/48)
tan (s - t) = 7/6 / (33/48)
tan (s - t) = 7/6 * 48/33
tan (s - t) = 56/33
So, the correct answer is tan (s - t) = 56/33, not 56/53.