Tutor profile: Grace G.
Evaluate sin(75°). Give an exact answer.
We can use trigonometric identities and the unit circle to solve this problem. We know that sin(a+b) = sin(a)cos(b) + sin(b)cos(a). If we let a = 30° and b = 45°, then sin(75)=sin(30+45)=sin(30)cos(45) + sin(45)cos(30). Using the unit circle, we know that sin(30) = 1/2, cos(45) = sqrt(2)/2, sin(45) = sqrt(2)/2, and cos(30) = sqrt(3)/2. Plugging these into our formula, we get sin(75) = (1/2)(sqrt(2)/2) + (sqrt(2)/2)(sqrt(3)/2) = sqrt(2)/4 + sqrt(6)/4. We can combine our answer into one fraction if we'd like, which is [sqrt(2)+sqrt(6)/4].
Use the product rule to find the derivative of the function h(t) = (t^2+3)(sin(t)).
The product rule says that (fg)' = f'g + g'f. If we make our f function f(t) = t^2+3 and our g function g(t)=sin(t), then f'(t)=2t and g'(t)=cos(t). Therefore, the derivative of h(t) is h'(t)=(2t)(sin(t)) + cos(t)(t^2+3).
Athena is practicing for a violin audition. She must practice for a total of 200 hours to sufficiently prepare for her audition. Athena has already practiced for 15 hours, and is able to practice for 2.5 hours each day before her audition. Write and solve an equation to figure out how many more days she needs to practice.
Since Athena has already practiced the violin for 15 hours, she only needs to practice for 200-15=185 more hours. If x represents the number of days Athena needs to practice, the equation is 185=2.5x. Divide both sides of the equation by 2.5 to isolate x. 185/2.5=x. Therefore, x=74. So, Athena must practice for 74 more days before her audition.
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