Evaluate: tan(arcsin(8/17)+ arctan(4/3))
I understand that I have to make triangles out of the values given-- with 15 being the other side of the 17 and 8 triangle, and 5 being the other side of the 4 and 3 triangle. I'm not sure where to go on from there, though.
Yeah, you have to draw two different triangles for them. It's easier to solve this if you draw them.
Anyway, we let arcsin(8/17) be equal to some angle, A. And we let arctan(4/3) be equal to some angle, B. In your drawing of two triangles, label these angles as A and B.
We can rewrite the expression tan(arcsin(8/17)+ arctan(4/3)) as:
tan(A + B)
Using the formula for sum of tangents, we'll have its expanded for:
( tan(A) + tan(B) ) / ( 1 - tan(A)tan(B) )
Since you have your drawing of triangles, you now put values for each. From the drawing, we know that
tan A = 8/15, and
tan B = 4/3
substituting,
= ( 8/15 + 4/3 ) / ( 1 - 8/15 * 4/3 )
= 84/13
hope this helps~ `u`
Ah, that makes sense! I always forget the trig addition formulas, thank you!
To evaluate the expression tan(arcsin(8/17) + arctan(4/3)), we can start by using the trigonometric identity that states:
tan(x + y) = (tan(x) + tan(y))/(1 - tan(x)*tan(y))
In this case, let x = arcsin(8/17) and y = arctan(4/3).
First, let's find the values of arcsin(8/17) and arctan(4/3):
arcsin(8/17) is the angle whose sine is 8/17. You correctly identified that this forms a right triangle with sides 8, 15, and 17. Therefore, the angle is opposite the side with length 8. Using the inverse sine function, we can find that arcsin(8/17) is approximately 0.472 radians (or approximately 27.09 degrees).
arctan(4/3) is the angle whose tangent is 4/3. This forms a right triangle with sides 3, 4, and 5. Therefore, the angle is opposite the side with length 4. Using the inverse tangent function, we can find that arctan(4/3) is approximately 0.93 radians (or approximately 53.13 degrees).
Now we have the values of x and y:
x = 0.472 radians
y = 0.93 radians
Substituting these values into the trigonometric identity, we get:
tan(0.472 + 0.93) = (tan(0.472) + tan(0.93))/(1 - tan(0.472)*tan(0.93))
Now, we can evaluate the individual tangents:
tan(0.472) ≈ 0.4973
tan(0.93) ≈ 1.7127
Plugging these values back into the equation, we have:
tan(0.472 + 0.93) ≈ (0.4973 + 1.7127) / (1 - 0.4973*1.7127)
Simplifying further:
tan(0.472 + 0.93) ≈ 2.21 / 0.1474
Finally, we can divide these values:
tan(0.472 + 0.93) ≈ 14.998
Therefore, evaluating the expression tan(arcsin(8/17) + arctan(4/3)) gives an approximate value of 14.998.