Given: sin37 = 6/10
use identities and calculate the following without using a calculator
Two things:
--- sin 37° ≠ 6/10
--- there are no "following"
sin37° is very close to 0.6, so cut 'em some slack.
The 6-8-10 triangle has one angle of 36.87°
I assume they want other trig functions using that triangle.
To calculate the following without using a calculator, we will use trigonometric identities and ratios. Let's break down the problem step by step:
1. Given: sin(37) = 6/10.
This means that in a right triangle, where the angle opposite to the side we are interested in is 37 degrees, the ratio of the length of the side opposite to the hypotenuse is 6/10.
2. To find the other trigonometric ratios (cosine, tangent, cosecant, secant, and cotangent), we can use the Pythagorean identity:
sin^2(x) + cos^2(x) = 1.
3. Using the Pythagorean identity, we can find the cosine of 37 degrees:
cos^2(37) = 1 - sin^2(37).
cos^2(37) = 1 - (6/10)^2.
cos^2(37) = 1 - 36/100.
cos^2(37) = 100/100 - 36/100.
cos^2(37) = 64/100.
cos(37) = sqrt(64/100).
cos(37) = 8/10.
cos(37) = 4/5.
4. Now, we can find the tangent of 37 degrees:
tan(37) = sin(37) / cos(37).
tan(37) = (6/10) / (4/5).
tan(37) = (6/10) * (5/4).
tan(37) = 30/40.
tan(37) = 3/4.
5. Continuing, we can find the cosecant of 37 degrees:
csc(37) = 1 / sin(37).
csc(37) = 1 / (6/10).
csc(37) = 1 * (10/6).
csc(37) = 10/6.
csc(37) = 5/3.
6. Similarly, we can find the secant of 37 degrees:
sec(37) = 1 / cos(37).
sec(37) = 1 / (4/5).
sec(37) = 1 * (5/4).
sec(37) = 5/4.
7. Lastly, we can find the cotangent of 37 degrees:
cot(37) = 1 / tan(37).
cot(37) = 1 / (3/4).
cot(37) = 1 * (4/3).
cot(37) = 4/3.
Therefore, without using a calculator, we have calculated the following trigonometric ratios using the given value of sin(37) = 6/10:
cos(37) = 4/5,
tan(37) = 3/4,
csc(37) = 5/3,
sec(37) = 5/4,
cot(37) = 4/3.