Write expressions for other five trigonometric ratios for each acute angle:
a. csc(theta 0) = a/b
b. sec(beta B) = x+1/x-1, x not equal 1
I will do the first one, then you do the second one
cscØ = a/b
or
sinØ = b/a
so make a sketch of a right-angled triangle where the base angle is Ø, its opposite side is b and the hypotenuse is a
let the adjacent side be c
b^2 + c^2 = a^2
c^2 = a^2 - b^2
c = √(a^2 - b^2)
so
sinØ = b/a ---------- cscØ = a/b (that was given)
cosØ = √(a^2-b^2)/a --- secØ = a/√(a^2-b^2)
tanØ = b/√(a^2-b^2) --- cotØ = √(a^2-b^2)/b
a. For the given acute angle theta (θ), the other five trigonometric ratios are as follows:
1. sec(θ): reciprocal of the cosine of theta.
sec(θ) = 1/cos(θ)
2. cot(θ): reciprocal of the tangent of theta.
cot(θ) = 1/tan(θ)
3. cosec(θ): reciprocal of the sine of theta.
cosec(θ) = 1/sin(θ)
4. tan(θ): quotient of the sine of theta divided by the cosine of theta.
tan(θ) = sin(θ)/cos(θ)
5. cos(θ): quotient of the adjacent side divided by the hypotenuse in a right triangle.
cos(θ) = adjacent/hypotenuse
b. For the given acute angle beta (B), the other five trigonometric ratios are as follows:
1. csc(B): reciprocal of the sine of beta.
csc(B) = 1/sin(B)
2. sec(B): reciprocal of the cosine of beta.
sec(B) = 1/cos(B)
3. cot(B): reciprocal of the tangent of beta.
cot(B) = 1/tan(B)
4. tan(B): quotient of the sine of beta divided by the cosine of beta.
tan(B) = sin(B)/cos(B)
5. cos(B): quotient of the adjacent side divided by the hypotenuse in a right triangle.
cos(B) = adjacent/hypotenuse
Note: Please make sure to substitute specific values of theta and beta in respective trigonometric functions to get the actual numerical values.
To find the expressions for the remaining five trigonometric ratios for each acute angle, we can use the definitions of these ratios and the given expressions for csc(theta) and sec(beta).
a. csc(theta) = a/b
The remaining trigonometric ratios are:
- sin(theta) = 1/csc(theta) = b/a
- cos(theta) = √(1 - sin^2(theta)) = √(1 - (b/a)^2)
- tan(theta) = sin(theta)/cos(theta) = (b/a) / √(1 - (b/a)^2)
- cot(theta) = 1/tan(theta) = √(1 - (b/a)^2) / (b/a)
- sec(theta) = 1/cos(theta) = 1 / √(1 - (b/a)^2)
b. sec(beta) = (x+1)/(x-1), with x ≠ 1
The remaining trigonometric ratios are:
- cos(beta) = 1/sec(beta) = (x-1)/(x+1)
- sin(beta) = √(1 - cos^2(beta)) = √(1 - ((x-1)/(x+1))^2)
- tan(beta) = sin(beta)/cos(beta) = √(1 - ((x-1)/(x+1))^2) / ((x-1)/(x+1))
- cot(beta) = 1/tan(beta) = ((x+1)/(x-1)) / √(1 - ((x-1)/(x+1))^2)
- csc(beta) = 1/sin(beta) = 1 / √(1 - ((x-1)/(x+1))^2)