if tan theta = 3/4 and 180 degrees ≤ 0 ≤ 270 degrees determine with the aid of a complete sketch
1. sin theta
2. cos(180 +theta)
3. angle theta
By "a complete sketch" they very likely want a right-angled triangle.
since tanØ = opposite/adjacent = 3/4 and the angle is in the third quadrant
y = -3 and x = -4
r^2 = 9 + 16 = 25
r = 5
sin Ø = -4/5
cos(Ø + 180)
= cosØcos180 - sinØsin180
= (cosØ)(-1) - 0
= - cosØ , you should know that identity
= -(-4/5)
= 4/5
Ø = 180 + tan^-1 (3/4)
= appr 216.9°
To determine the values of sin theta, cos(180 + theta), and the angle theta, let's go step by step:
Step 1: Determining sin theta.
Given that tan theta = 3/4, we can find sin theta using the following identity:
sin theta = tan theta / √(1 + tan^2 theta)
Substituting the given tan theta value:
sin theta = (3/4) / √(1 + (3/4)^2)
sin theta = (3/4) / √(1 + 9/16)
sin theta = (3/4) / √(16/16 + 9/16)
sin theta = (3/4) / √(25/16)
sin theta = (3/4) / (5/4)
sin theta = 3/5
So, sin theta = 3/5.
Step 2: Determining cos(180 + theta).
Using the identities cos(180 + theta) = -cos theta and cos theta = 1/sqrt(1 + tan^2 theta), we can find cos(180 + theta) as follows:
cos(180 + theta) = -cos theta
cos(180 + theta) = -1/sqrt(1 + tan^2 theta)
cos(180 + theta) = -(1/√(1 + (3/4)^2))
cos(180 + theta) = -(1/√(1 + 9/16))
cos(180 + theta) = -(1/√(16/16 + 9/16))
cos(180 + theta) = -(1/√(25/16))
cos(180 + theta) = -(1/(5/4))
cos(180 + theta) = -(4/5)
cos(180 + theta) = -4/5
So, cos(180 + theta) = -4/5.
Step 3: Determining the angle theta.
Given that 180 degrees ≤ theta ≤ 270 degrees, and using the inverse tangent function, we can find that:
theta = tan^(-1)(3/4)
theta ≈ 36.87 degrees
So, the angle theta is approximately 36.87 degrees.
In summary:
1. sin theta = 3/5
2. cos(180 + theta) = -4/5
3. Angle theta ≈ 36.87 degrees
To find the values of the trigonometric functions and the angle from the given information, we can use the definitions and properties of trigonometric functions. Here's how you can find the answers:
1. sin theta:
Since we are given the value of tan theta, we can use the identity "tan theta = sin theta / cos theta" to find sin theta.
Given: tan theta = 3/4
Using the identity, we have:
3/4 = sin theta / cos theta
To find sin theta, we need the value of cos theta. Since we don't have that information directly, we can use the Pythagorean identity "sin^2 theta + cos^2 theta = 1" to solve for cos theta.
We know that theta is in the third quadrant (180 degrees ≤ theta ≤ 270 degrees), where both sine and cosine are negative. Therefore, we should use the negative values for sin theta and cos theta.
Using the Pythagorean identity, we have:
sin^2 theta + cos^2 theta = 1
sin^2 theta + (-cos theta)^2 = 1
sin^2 theta + cos^2 theta = 1
sin^2 theta + cos^2 theta = 1
Plugging in the given value of tan theta and the negative sign for both sin theta and cos theta:
(3/4)^2 + (-cos theta)^2 = 1
9/16 + cos^2 theta = 1
cos^2 theta = 16/16 - 9/16
cos^2 theta = 7/16
Since cos theta is negative (third quadrant) and we have cos^2 theta, we take the negative square root:
cos theta = -√(7)/4
Now, we can solve for sin theta:
sin theta = (3/4) / (-√(7)/4)
sin theta = -3/√(7)
So, sin theta = -3/√(7).
2. cos(180 + theta):
We know that cos(a + b) = cos a * cos b - sin a * sin b. By applying this identity, we can find cos(180 + theta):
cos(180 + theta) = cos 180 * cos theta - sin 180 * sin theta
cos 180 = -1 and sin 180 = 0, so:
cos(180 + theta) = -1 * cos theta - 0 * sin theta
cos(180 + theta) = - cos theta
Therefore, cos(180 + theta) = -(-√7/4) = √7/4.
3. Angle theta:
Given: tan theta = 3/4
We can find angle theta by taking the inverse tangent (also known as arctan) of the given tangent value.
Using a scientific calculator, find the inverse tangent (arctan) of 3/4:
theta = arctan(3/4)
theta ≈ 36.87 degrees
So, angle theta is approximately 36.87 degrees.
To create a complete sketch, you can use a unit circle or draw a coordinate plane and plot the points corresponding to theta and the trigonometric values you found.