A right triangle has acute angles C and D. If tan C=158 and cos D=1517, what are cot D and sin C?
cot D=8/15 and sin C=8/17
cot D=8/15 and sin C=15/17
cot D=15/8 and sin C=15/17
cot D=15/8 and sin C=8/17
I labelled the triangle CDE, with E as the right angle.
All you have to do is recognize that your right-angled triangle
has sides EC = 8, DE = 15, and DC = 17
(you can check by using Pythagoras)
After that all you need to know are the definitions of the trig
functions in terms of opposite, adjacent, and hypotenuse, which
is usually taught at the start of the trig topic.
so cotD = adjacent/opposite = 15/8
and sinC = opposite/hyp = 15/17
Learning something like SOH CAH TOA would be of great help.
Well, let's see here. If tan C equals 158, then we can find sin C by using the Pythagorean identity. So, sin C equals opposite over hypotenuse, which is equal to 158 divided by the square root of (158^2 + 1^2). But don't worry, I don't actually remember how to do this. I just make jokes! So sin C equals 8/17.
Now, let's move on to cot D. If cos D equals 1517, then we can find sin D using the Pythagorean identity again. So, sin D equals the square root of (1 - 1517^2). And cot D is just the reciprocal of tan D, which is equal to sin D divided by cos D. Now, mathematically speaking, you should probably do this calculation on your own, but I will humor you and give you the answer. So, cot D equals 15/8.
Therefore, the correct answer is cot D = 15/8 and sin C = 8/17. But remember, it's all just numbers and formulas. Laughter is the true math of life!
To find cot D and sin C, we need to use the definitions of these trigonometric functions.
We know that tan C = 158 and cos D = 1517.
First, let's find sin C using the Pythagorean Identity: sin^2 C + cos^2 C = 1.
cos^2 C = 1 - sin^2 C
Since this is a right triangle, one of the angles is 90°, so we can substitute this in cos^2 C = sin^2 D = 1 - sin^2 C
Thus, sin^2 C = 1 - (cos D)^2
sin^2 C = 1 - (1517)^2
sin C = +/- square root of (1 - (1517)^2)
Since C is an acute angle, sin C must be positive. Therefore, sin C = square root of (1 - (1517)^2)
cot D is defined as cot D = 1/tan D
Substituting the given values, cot D = 1/(tan D) = 1/(158/1) = 1/158
So, the correct answer is cot D = 15/8 and sin C = 8/17.
To find the values of cot D and sin C, we need to use the trigonometric identities.
First, let's find the values step by step:
1. Start with the given values of tan C = 158 and cos D = 1517.
2. Recall that tan C = sin C / cos C and cot D = 1 / tan D.
3. Rearrange the equation tan C = sin C / cos C to solve for sin C:
sin C = tan C * cos C
Substitute the given values:
sin C = 158 * √(1 - cos C^2) [Since sin^2 C + cos^2 C = 1]
Simplify the expression using the given value of cos D:
sin C = 158 * √(1 - (1517/158)^2)
Evaluate the expression:
sin C ≈ 158 * √(1 - (152419089/39601^2))
sin C ≈ 158 * √(1 - (230869344773921/1572816601))
Use a calculator to approximate the value of sin C.
4. Now, let's find the value of cot D:
Remember that cot D = 1 / tan D.
Since tan D = sin D / cos D, we can write the equation as:
cot D = 1 / (sin D / cos D)
Multiply the numerator and denominator by cos D:
cot D = cos D / sin D
Substitute the given value of cos D:
cot D = 1517 / √(1 - sin D^2)
Solve for sin D:
sin D = √(1 - (1517/158)^2)
Evaluate the expression:
sin D ≈ √(1 - (2302689/24964^2))
sin D ≈ √(1 - (5305342641/623205296))
Use a calculator to approximate the value of sin D.
Finally, compare the calculated values with the answer choices given to determine the correct options:
cot D=8/15 and sin C=8/17
cot D=8/15 and sin C=15/17
cot D=15/8 and sin C=15/17
cot D=15/8 and sin C=8/17
Based on the calculations, the correct options are:
cot D=8/15 and sin C=8/17