The point (1/3,1/4) lies on the terminal side of an angle.
Find the exact value of the six trig functions, and explain which functions are reciprocal functions to each other.
where in the good mother of jebus did the rest of the answers go after the son like nahhhh i needed that to survive precalcc omlllll
To find the exact values of the six trig functions (sine, cosine, tangent, cosecant, secant, and cotangent) for the angle whose terminal side contains the point (1/3, 1/4), we can use the following steps:
Step 1: Determine the values of the triangle's sides:
The coordinates of the point (1/3, 1/4) represent a right triangle in the first quadrant. The length of the side adjacent to the angle can be found by subtracting the x-coordinate (1/3) from 1: 1 - 1/3 = 2/3. The length of the side opposite to the angle can be obtained by subtracting the y-coordinate (1/4) from 1: 1 - 1/4 = 3/4. Then, we can use the Pythagorean theorem to find the hypotenuse (c): c^2 = (2/3)^2 + (3/4)^2.
Simplifying, we get:
c^2 = 4/9 + 9/16
c^2 = (64 + 81) / (9 * 16)
c^2 = 145 / (9 * 16)
c^2 = 145 / 144
c = √(145) / 12
Step 2: Calculate the six trig function values:
a) Sine (sinθ) = opposite/hypotenuse = (3/4) / (√145 / 12) = 3√145 / 4 * 12 = 3√145 / 48 = √145 / 16
b) Cosine (cosθ) = adjacent/hypotenuse = (2/3) / (√145 / 12) = 2√145 / 3 * 12 = 2√145 / 36 = √145 / 18
c) Tangent (tanθ) = opposite/adjacent = (3/4) / (2/3) = (3/4) * (3/2) = 9/8
d) Cosecant (cscθ) = 1/sinθ = 1 / (√145 / 16) = 16 / √145
e) Secant (secθ) = 1/cosθ = 1 / (√145 / 18) = 18 / √145
f) Cotangent (cotθ) = 1/tanθ = 1 / (9/8) = 8/9
Step 3: Identify reciprocal functions:
Reciprocal functions are pairs of trigonometric functions that are multiplicative inverses of each other. In this case, the reciprocal functions are:
- Sine (sinθ) and Cosecant (cscθ): sinθ * cscθ = 1
- Cosine (cosθ) and Secant (secθ): cosθ * secθ = 1
- Tangent (tanθ) and Cotangent (cotθ): tanθ * cotθ = 1
To find the exact values of the six trigonometric functions (sine, cosine, tangent, cosecant, secant, and cotangent) for the given point (1/3, 1/4) on the terminal side of an angle, we can use the concept of right triangles.
Step 1: Draw a right triangle using the given point (1/3, 1/4) as one of its vertices, with the hypotenuse on the positive x-axis and extending towards the origin.
Step 2: Label the sides of the triangle. The horizontal side adjacent to the angle is the x-coordinate of the point, which is 1/3. The vertical side opposite the angle is the y-coordinate of the point, which is 1/4. The hypotenuse of the triangle is the distance from the origin to the point, which can be found using the Pythagorean theorem.
Step 3: Calculate the length of the hypotenuse. The length of the hypotenuse squared is the sum of the squares of the other two sides:
(1/3)^2 + (1/4)^2 = 1/9 + 1/16 = (16 + 9)/144 = 25/144
So, the length of the hypotenuse is the square root of 25/144. Taking the positive square root, we get:
√(25/144) = 5/12
Step 4: Now, we can calculate the six trigonometric functions based on the sides of the triangle:
- Sine (sin): sin(θ) = opposite/hypotenuse = (1/4) / (5/12) = 1/4 * 12/5 = 3/10
- Cosine (cos): cos(θ) = adjacent/hypotenuse = (1/3) / (5/12) = 1/3 * 12/5 = 4/5
- Tangent (tan): tan(θ) = opposite/adjacent = (1/4) / (1/3) = 1/4 * 3/1 = 3/4
- Cosecant (csc): csc(θ) = 1/sin(θ) = 1/(3/10) = 10/3
- Secant (sec): sec(θ) = 1/cos(θ) = 1/(4/5) = 5/4
- Cotangent (cot): cot(θ) = 1/tan(θ) = 1/(3/4) = 4/3
So, the exact values of the six trigonometric functions for the angle with the terminal point (1/3, 1/4) are:
sin(θ) = 3/10
cos(θ) = 4/5
tan(θ) = 3/4
csc(θ) = 10/3
sec(θ) = 5/4
cot(θ) = 4/3
Regarding reciprocal functions, the reciprocal of a trigonometric function is found by taking the reciprocal of the corresponding trigonometric ratio. The following pairs of functions are reciprocals of each other:
- Sine and Cosecant: sin(θ) and csc(θ) are reciprocals of each other.
- Cosine and Secant: cos(θ) and sec(θ) are reciprocals of each other.
- Tangent and Cotangent: tan(θ) and cot(θ) are reciprocals of each other.
Reciprocal functions have a special relationship where multiplying them together results in 1. For example, sin(θ) * csc(θ) = 1, cos(θ) * sec(θ) = 1, and tan(θ) * cot(θ) = 1.
Recall that
sinθ = y/r
cosθ = x/r
tanθ = y/x
and the others are their reciprocals.
So, since
x = 1/3 = 4/12
y = 1/4 = 3/12
You have the makings of a 3-4-5 triangle, with r = 5/12
Now you can just read off the functions
sinθ = 4/5
cosθ = 3/5
tanθ = 3/4
and so on...