1- If csc θ = 5 and θ is an acute angle, find:
a. sin θ
b. cos θ
2- Use trig identities to transform the expression (sec θ)/(csc θ) into a single trig function. Assume θ is acute.
7. Using the table of values, find the value of θ in radians (assume θ is acute):
a. cot θ = square root of 3
b. sec θ = 2
c. csc θ = 2/ square root of3
d. tan θ = 1
#1. sinθ = 1/cscθ
cos^2θ + sin^2θ = 1
#2. secθ/cscθ = 1/cosθ * sinθ = tanθ
#3. Review the special angles: 0, π/6, π/4, π/3, π/2
1- If csc θ = 5 and θ is an acute angle, we can find sin θ and cos θ using the following steps:
a. sin θ = 1 / csc θ
sin θ = 1 / 5
b. cos θ = √(1 - sin^2θ)
cos θ = √(1 - (1/5)^2)
cos θ = √(1 - 1/25)
cos θ = √(24/25)
cos θ = √24 / 5
2- To transform the expression (sec θ)/(csc θ) into a single trig function, we use the reciprocal identities:
(sec θ)/(csc θ) = (1/cos θ)/(1/sin θ)
= sin θ / cos θ
= tan θ
Therefore, the expression (sec θ)/(csc θ) is equivalent to tan θ.
7- To find the value of θ in radians for each trigonometric function:
a. cot θ = √3
Use the inverse cotangent function (cot^-1) to find the value of θ:
θ = cot^-1(√3)
≈ 0.7854 radians
b. sec θ = 2
Use the inverse secant function (sec^-1) to find the value of θ:
θ = sec^-1(2)
≈ 1.0472 radians
c. csc θ = 2/√3
Use the inverse cosecant function (csc^-1) to find the value of θ:
θ ≈ csc^-1(2/√3)
≈ 1.0472 radians
d. tan θ = 1
Use the inverse tangent function (tan^-1) to find the value of θ:
θ = tan^-1(1)
≈ 0.7854 radians
1a. To find sin θ, we can use the reciprocal identity of csc θ. Since csc θ = 1/sin θ, we can rearrange the equation to get sin θ = 1/csc θ.
Given that csc θ = 5, we can substitute the value into the equation, giving sin θ = 1/5.
1b. To find cos θ, we can use the Pythagorean identity. Since sin θ = 1/csc θ, we can apply the Pythagorean identity sin² θ + cos² θ = 1 to solve for cos θ.
Rearranging the equation, we get cos θ = √(1 - sin² θ).
Plugging in the known value of sin θ = 1/5, we can find cos θ ≈ √(1 - 1/25) ≈ √(24/25) ≈ √24/5.
2. To transform the expression (sec θ)/(csc θ) into a single trig function, we can simplify using the reciprocal identities.
Recall that sec θ = 1/cos θ and csc θ = 1/sin θ.
So, (sec θ)/(csc θ) can be written as (1/cos θ)/(1/sin θ).
Simplifying further, we have (sin θ)/(cos θ) = tan θ.
Therefore, the expression (sec θ)/(csc θ) can be transformed into tan θ.
3a. To find the value of θ when cot θ = √3, we can use the reciprocal identity of cot θ. Since cot θ = 1/tan θ, we can rearrange the equation to get tan θ = 1/cot θ.
Plugging in the given value of cot θ = √3, we can find tan θ = 1/(√3) = √3/3.
We can use inverse trigonometric functions to find the value of θ, which is approximately 0.615 radians.
3b. To find the value of θ when sec θ = 2, we can use the reciprocal identity of sec θ. Since sec θ = 1/cos θ, we can rearrange the equation to get cos θ = 1/sec θ.
Plugging in the given value of sec θ = 2, we can find cos θ = 1/2.
We can use inverse trigonometric functions to find the value of θ, which is approximately 1.047 radians.
3c. To find the value of θ when csc θ = 2/√3, we can use the reciprocal identity of csc θ. Since csc θ = 1/sin θ, we can rearrange the equation to get sin θ = 1/csc θ.
Plugging in the given value of csc θ = 2/√3, we can find sin θ = 1/(2/√3) = √3/2.
We can use inverse trigonometric functions to find the value of θ, which is approximately 1.047 radians.
3d. To find the value of θ when tan θ = 1, we can use the inverse tangent function.
Using the inverse tangent function tan⁻¹(1) or atan(1), we can find the value of θ, which is approximately 0.785 radians (or π/4 radians).