if tan theta= 1/2, find the exact values of the five remaining trigonometric functions for theta, which is acute
To find the exact values of the five remaining trigonometric functions for theta, given that tan theta is equal to 1/2, we can use the following steps:
Step 1: Use the given ratio to find the actual value of the angle theta. Since we know that tan theta is equal to 1/2, we can determine that theta must be one of the acute angles that has a tangent of 1/2.
Step 2: Find the values of the other trigonometric functions (sin, cos, cot, sec, and csc) using the values of tan theta and theta that we obtained in Step 1.
Let's go through the steps in detail:
Step 1: Finding the value of theta
To determine the value of theta, we need to find which acute angle has a tangent of 1/2. The acute angle that satisfies this condition is 30 degrees (or pi/6 radians). Another acute angle with a tangent of 1/2 is 150 degrees (or 5pi/6 radians).
So, theta can be either 30 degrees or 150 degrees.
Step 2: Finding the values of the other trigonometric functions
To find the values of the other trigonometric functions, we can use the relationships between them.
For theta = 30 degrees (or pi/6 radians), the five remaining trigonometric functions are:
1. sin theta: Since sin theta = opposite/hypotenuse, and in a 30-60-90 triangle, the opposite side is 1 and the hypotenuse is 2 (because the ratio of the sides in a 30-60-90 triangle is 1:√3:2), sin theta = 1/2.
2. cos theta: Since cos theta = adjacent/hypotenuse, and in a 30-60-90 triangle, the adjacent side is √3 (because the ratio of the sides is 1:√3:2), and the hypotenuse is 2, cos theta = √3/2.
3. cot theta: Since cot theta = 1/tan theta, and tan theta = 1/2, cot theta = 2.
4. sec theta: Since sec theta = 1/cos theta, and cos theta = √3/2, sec theta = 2/√3.
5. csc theta: Since csc theta = 1/sin theta, and sin theta = 1/2, csc theta = 2.
For theta = 150 degrees (or 5pi/6 radians), the values of the remaining trigonometric functions will be the negative of the values obtained in Step 2 for theta = 30 degrees.
So, for theta = 150 degrees (or 5pi/6 radians), the five remaining trigonometric functions are:
1. sin theta = -1/2
2. cos theta = -√3/2
3. cot theta = -2
4. sec theta = -2/√3
5. csc theta = -2
Therefore, the exact values of the five remaining trigonometric functions for theta, given that tan theta is equal to 1/2, for both acute angles 30 degrees and 150 degrees are:
For 30 degrees:
sin theta = 1/2
cos theta = √3/2
cot theta = 2
sec theta = 2/√3
csc theta = 2
For 150 degrees:
sin theta = -1/2
cos theta = -√3/2
cot theta = -2
sec theta = -2/√3
csc theta = -2
To find the exact values of the remaining trigonometric functions for theta, we can use the properties of the right triangle.
Given that tan(theta) = 1/2, we can draw a right triangle where the ratio of the opposite side to the adjacent side is 1/2. Let's assume the opposite side is x and the adjacent side is 2x.
Using the Pythagorean theorem, we have:
(x)^2 + (2x)^2 = (hypotenuse)^2
Simplifying the equation, we get:
5x^2 = (hypotenuse)^2
To find the hypotenuse, we take the square root of both sides:
sqrt(5x^2) = hypotenuse
sqrt(5)x = hypotenuse
Now let's find the values of the remaining trigonometric functions:
1. sin(theta) = opposite/hypotenuse = x/sqrt(5)x = 1/sqrt(5)
2. cos(theta) = adjacent/hypotenuse = 2x/sqrt(5)x = 2/sqrt(5) = 2sqrt(5)/5
3. csc(theta) = 1/sin(theta) = sqrt(5)/1 = sqrt(5)
4. sec(theta) = 1/cos(theta) = sqrt(5)/2
5. cot(theta) = 1/tan(theta) = 1/(1/2) = 2
just draw the triangle. The sides are 1,2,√5
Now just read off the trig ratios.