Suppose
a
represents some number of radians where
0
<
a
<
π
2
, and
cos
(
a
)
=
0.7
and
sin
(
a
)
=
0.72
. Determine the values of the following expressions.
cos
(
−
a
)
=
sin
(
−
a
)
=
Hint: it might help to start by drawing a diagram of two angles: one with a measure of
a
radians, and one with a measure of
−
a
radians.
your bizarre text formatting makes my eyes bleed.
Well, since cos(a) = 0.7 and sin(a) = 0.72, we know that a is a special angle. And because a is positive, we can say that it lies in the first quadrant. So, let's draw a little diagram!
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Okay, let's label the positive x-axis as "x" and the positive y-axis as "y". Now, let's label the side lengths of the right triangle formed in the first quadrant. The adjacent side to angle a is "x", and the opposite side is "y". The hypotenuse is then the square root of x^2 + y^2.
Now, we can use the given information: cos(a) = 0.7 and sin(a) = 0.72. We know that cos(a) = x / hypotenuse, so we can replace cos(a) with x / square root of x^2 + y^2. Similarly, sin(a) = y / hypotenuse, so we can replace sin(a) with y / square root of x^2 + y^2.
Now, let's solve for x and y:
x / √(x^2 + y^2) = 0.7
y / √(x^2 + y^2) = 0.72
Now, let's square both sides of the equations:
x^2 / (x^2 + y^2) = 0.49
y^2 / (x^2 + y^2) = 0.5184
Now, let's subtract both equations to eliminate x:
x^2 / (x^2 + y^2) - y^2 / (x^2 + y^2) = 0.49 - 0.5184
(x^2 - y^2) / (x^2 + y^2) = -0.0284
Now, let's substitute a variable to make the equation more manageable:
Let z = x^2 + y^2, then we have:
(x^2 - y^2) / z = -0.0284
x^2 - y^2 = -0.0284z
Since x^2 + y^2 = z, we can simplify the equation further:
x^2 - y^2 = -0.0284(x^2 + y^2)
Expanding the equation:
x^2 - y^2 = -0.0284x^2 - 0.0284y^2
Rearranging the equation:
0.9716x^2 - 0.9716y^2 = 0
What have we done? We've derived the equation of an ellipse! But wait, we're solving for cos(-a) and sin(-a), not trying to find the equation of an ellipse.
I'm sorry, but it seems I've gotten carried away in my calculations. Let's do things the easier way by actually applying the definition of cos and sin for an angle in the first quadrant.
cos(-a) is simply the cosine of the negative of angle a. Since we know that cos(a) = 0.7, we know that cos(-a) = cos(a) = 0.7.
Similarly, sin(-a) is the sine of the negative of angle a. Since we know that sin(a) = 0.72, we know that sin(-a) = -sin(a) = -0.72.
So, cos(-a) = 0.7 and sin(-a) = -0.72.
Hope that helps!
To determine the values of cos(-a) and sin(-a), let's start by understanding the relationship between these trigonometric functions and the angle -a.
1. Drawing a diagram:
Draw a coordinate plane and mark an angle "a" in the first quadrant. The point on the unit circle corresponding to this angle will have coordinates (cos(a), sin(a)) = (0.7, 0.72).
Next, draw an angle "-a" in the fourth quadrant. The point on the unit circle corresponding to this angle will have coordinates (-cos(a), -sin(a)) = (-0.7, -0.72).
2. Determine the values of cos(-a) and sin(-a):
From the diagram, we can see that cos(-a) is equal to the x-coordinate of the point (-0.7, -0.72). Therefore, cos(-a) = -0.7.
Similarly, sin(-a) is equal to the y-coordinate of the point (-0.7, -0.72). Therefore, sin(-a) = -0.72.
So, the values of the expressions are:
cos(-a) = -0.7
sin(-a) = -0.72
To determine the values of cos(-a) and sin(-a), let's start by understanding the relationship between the trigonometric functions in the unit circle.
The unit circle is a circle with a radius of 1 centered at the origin (0, 0) in a coordinate plane. It is often used to represent angles in trigonometry. Each point on the unit circle represents an angle and its corresponding trigonometric function values.
Given that cos(a) = 0.7 and sin(a) = 0.72, we can interpret this as follows:
- The x-coordinate of the point on the unit circle corresponding to angle a is 0.7 (cos(a)).
- The y-coordinate of the point on the unit circle corresponding to angle a is 0.72 (sin(a)).
To determine cos(-a), we need to find the x-coordinate of the point on the unit circle corresponding to angle -a. Since angle -a is the reflection of angle a about the x-axis, its x-coordinate will be the same as that of angle a.
Therefore, cos(-a) = cos(a) = 0.7.
To determine sin(-a), we need to find the y-coordinate of the point on the unit circle corresponding to angle -a. Since angle -a is the reflection of angle a about the x-axis, its y-coordinate will be the negative of the y-coordinate of angle a.
Therefore, sin(-a) = -sin(a) = -0.72.
So, the values of the expressions are:
cos(-a) = 0.7
sin(-a) = -0.72