Tutor profile: Michelle N.
Show that 1 + (tan(x))^2 = (sec(x))^2.
We look at the left hand side first. Using trig identities, we know that tan(x) can be rewritten as sin(x)/cos(x). Substituting that back into 1 + (tan(x))^2, we get 1 + (sin(x) / cos(x))^2. We can rewrite this entire left hand side so that 1 + (sin(x) / cos(x))^2 = [(cos(x))^2+(sin(x))^2] / (cos(x))^2 by rewriting 1 as (cos(x))^2 / (cos(x))^2. Using trig identities again, we know that the numerator (cos(x))^2+(sin(x))^2 equals to 1. So [(cos(x))^2+(sin(x))^2] / (cos(x))^2 equals to 1 / (cos(x))^2. Again, using trig identities, we know that 1 / (cos(x))^2 = (sec(x))^2. Thus since (sec(x))^2 = (sec(x))^2, we have shown that 1 + (tan(x))^2 = (sec(x))^2.
272/x = 16. Find x.
We can rewrite this equation by multiplying both sides by x so that we get x(272/x) = (16)x. On the left hand side, the variable will cancel out, giving us 272 = (16)x. Since we want to find x, we want to get rid of the 16. To do that, we divide both sides by 16 so that we get 272/16 = (16x)/16. On the right hand side, the 16 cancels out, leaving us with x. On the left hand side, the 272/16 can be simplified to be 17 by checking to see if both numerator and denomerator can be divided by 2, 3, 4, 5, and so on. Done correctly, 272/16 simplifies to be 17 on the left hand side. So now, we have 17 = x.
The price of a water bottle decreased from $25 to $20. By how many percent did the price decrease?
We want to find the percent by which the price changed for the water bottle. To do this, we need to find the absolute difference in price change and divide that by the original price. To start, we take the absolute value of the difference of the price | price_final - price_initial | = |20 - 25| = 5. So we know that 5 is the amount by which the price changed and that 25 is the original price. So we divide 5 by 25 to get 5/25 = 1/5 = 0.20. Multiply this value by 100 and we get 20%. So the price of a water bottle decreased by 20%.
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