For the acute angle with sin theta=3/5, find cos theta, tan, cot, csc, sec.
Draw the triangle. Isn't it a 3,4,5 triangle?
For an acute angle θ, and sin(θ)=3/5, all the other five functions will be positive.
Draw a right-triangle and label one of the acute angles θ.
Label the length of the hypothenuse as 5, the side opposite θ as 3. Confirm that sin(θ) is 3/5.
Calculate the adjacent (remaining) side length by Pythagoras's theorem, namely L=√(5²-3²)=4. Label the length of the adjacent side 4.
Now from the fundamental definitions of the functions, determine the trigonometric ratios of each from:
sin(θ) = opposite/hypothenuse
cos(θ) = adjacent/hypothenuse
tan(θ) = opposite/adjacent
csc(θ) = hypothenuse/opposite
sec(θ) = hypothenuse/adjacent
cot(θ) = adjacent/opposite
To find the values of cos theta, tan theta, cot theta, csc theta, and sec theta, given that sin theta is 3/5, we can use the following trigonometric identities:
1. cos(theta) = sqrt(1 - sin^2(theta))
2. tan(theta) = sin(theta) / cos(theta)
3. cot(theta) = 1 / tan(theta)
4. csc(theta) = 1 / sin(theta)
5. sec(theta) = 1 / cos(theta)
Using these identities, let's find each of the values step by step:
Step 1: Finding cos(theta)
cos(theta) = sqrt(1 - sin^2(theta))
cos(theta) = sqrt(1 - (3/5)^2)
cos(theta) = sqrt(1 - 9/25)
cos(theta) = sqrt(16/25)
cos(theta) = 4/5
So, cos(theta) is equal to 4/5.
Step 2: Finding tan(theta)
tan(theta) = sin(theta) / cos(theta)
tan(theta) = (3/5) / (4/5)
tan(theta) = 3/4
So, tan(theta) is equal to 3/4.
Step 3: Finding cot(theta)
cot(theta) = 1 / tan(theta)
cot(theta) = 1 / (3/4)
cot(theta) = 4/3
So, cot(theta) is equal to 4/3.
Step 4: Finding csc(theta)
csc(theta) = 1 / sin(theta)
csc(theta) = 1 / (3/5)
csc(theta) = 5/3
So, csc(theta) is equal to 5/3.
Step 5: Finding sec(theta)
sec(theta) = 1 / cos(theta)
sec(theta) = 1 / (4/5)
sec(theta) = 5/4
So, sec(theta) is equal to 5/4.
In summary, for the given value of sin theta = 3/5, the values of cos(theta), tan(theta), cot(theta), csc(theta), and sec(theta) are as follows:
cos(theta) = 4/5
tan(theta) = 3/4
cot(theta) = 4/3
csc(theta) = 5/3
sec(theta) = 5/4