Trig Questions-
1. Write the algebraic expression which shows Cos((ArcSin(4/X)),
2. Angle If Csc(-Θ)=15/4?
2. - what quadrant is it in, find the sin(-Θ), Sin(Θ), Tan(90deg-Θ), Tan(Θ), csc(Θ), Sec(-Θ+360deg), Thanks
These are rather simple if you make drawings or sketches.
For instance on the first, if sin is 4/x
then the hypotenuse is x, the opposite side is 4, and the adjacaent side is sqrt(x^2-4). So now what is the cosine?
You can do this. We will be happy to critique your thinking.
1. To determine the algebraic expression for Cos((ArcSin(4/X)), we need to use the relationship between inverse trigonometric functions and their related trigonometric functions.
First, let's identify the relationship between arcsine (ArcSin) and cosine (Cos). The relation is given as:
ArcSin(x) = θ ---> Sin(θ) = x
In our case, ArcSin(4/X) = θ. Applying the relation, we have:
Sin(θ) = 4/X
To find Cos((ArcSin(4/X))), we will use the Pythagorean identity: Sin^2(θ) + Cos^2(θ) = 1.
Since we have Sin(θ) = 4/X, we can substitute it in the equation above:
(4/X)^2 + Cos^2(θ) = 1
To solve for the value of Cos(θ), we need to rearrange the equation:
Cos^2(θ) = 1 - (4/X)^2
Cos(θ) = ± √(1 - (4/X)^2)
Therefore, the algebraic expression for Cos((ArcSin(4/X))) is:
Cos((ArcSin(4/X))) = ± √(1 - (4/X)^2)
2. To find the angle (θ) when Csc(-θ) = 15/4, we need to determine the value of θ.
First, let's recall the relationship between cosecant (Csc) and sine (Sin):
Csc(θ) = 1/Sin(θ)
In our case, we have Csc(-θ) = 15/4. Applying the relationship, we get:
1/Sin(-θ) = 15/4
By cross-multiplication, we have:
4 = 15/4 * Sin(-θ)
Now, let's solve for Sin(-θ) by rearranging the equation:
Sin(-θ) = 4 * (4/15)
Sin(-θ) = 16/15
Since Sin(-θ) = Sin(θ) (since sine is an odd function), we can rewrite the equation as:
Sin(θ) = 16/15
To find the angle θ, we can use the arcsine (ArcSin) function:
ArcSin(16/15) = θ
Hence, the angle (θ) when Csc(-θ) = 15/4 is obtained by taking the arcsine of 16/15.
3. To determine the quadrant in which an angle -θ lies and find various trigonometric ratios for it, we need to consider the signs of the trigonometric functions in each quadrant.
Here's the breakdown:
-θ lies in the fourth quadrant.
In the fourth quadrant:
- Sin(θ) > 0
- Cos(θ) < 0
- Tan(θ) > 0
- Csc(θ) < 0
- Sec(θ) < 0
Given that, we can determine the specific values of these trigonometric functions:
- Sin(-θ) = -Sin(θ)
- Sin(-θ) = -(-Sin(θ)) = Sin(θ)
- Cos(-θ) = Cos(θ)
- Tan(90° - θ) = -Tan(θ)
- Tan(θ) remains unchanged.
- Tan(-θ) = -Tan(θ)
- Csc(θ) remains unchanged.
- Sec(-θ + 360°) = Sec(θ)
- Sec(-θ + 360°) = Sec(-θ)
Note: It is important to remember that the trigonometric functions may have different magnitudes depending on the specific angle value, but the signs are determined by the quadrant in which the angle lies.
Hope this explanation helps!