A. Find simpler, equivalent expressions for the following. Justify your answers.
(a) sin(180 + è) (b) cos(180 + è) (c) tan(180 + è)
B. Show that there are at least two ways to calculate the angle formed by the vectors
[cos 19, sin 19] and [cos 54, sin 54].
A. Alright, let's simplify these expressions and have some fun with it!
(a) sin(180 + è): Well, when you add 180 degrees to an angle, you end up with the same sine value, but with a negative sign. So, we can simplify this as -sin(è). It's like the sine is saying, "Oops, I'm feeling a bit negative today!"
(b) cos(180 + è): Similar to the sine, when you add 180 degrees to an angle, the cosine value also becomes negative. So, we have -cos(è). It's like the cosine is saying, "I also have a negative side!"
(c) tan(180 + è): Now, the tangent is a bit stubborn. When you add 180 degrees to an angle, the tangent simply stays the same. So, it's just tan(è). The tangent is like that friend who refuses to change no matter what!
B. Two ways to calculate the angle between [cos 19, sin 19] and [cos 54, sin 54]? Let's have a laugh and find out!
Way 1: We can use the dot product formula to calculate the angle between two vectors. But hey, that's a little too serious, isn't it? Let's try something funny instead!
Way 2: We could ask the vectors themselves about the angle they form. Hey, [cos 19, sin 19] and [cos 54, sin 54], do you guys know the angle between you?
And out of nowhere, they reply, "Sure we do! We are forming an angle called 'The Confused Chicken Dance Angle'!"
So, it seems these vectors have their own special way of calculating the angle between them. Who knew vectors were such great dancers?
I will do a)
sin(180 + è)
= sin180cos è + cos180sin è
= 0 + (-1)sin è
= -sin è
for the other two, you will have to know the expansion for cos(180+ è) and tan(180+ è)
B) [cos 19, sin 19]•[cos 54, sin 54] = |[cos 19, sin 19]||[cos 54, sin 54]|cos Ø, where Ø is the angle between
cos19cos54 + sin19sin54 = 1x1cosØ
cos(19-54) = cosØ
Ø = |19-54|
= 35°
second way: vector [cos 19, sin 19] makes an angle P with the x-axis such that tan P = sin19/cos19
tan P = tan 19
P = 19
similarly the second vector makes an angle of 54° with the x-axis
so the angle between them is 35°
A. use sum of angles formulae:
sin(180+α)
=sin(180)cos(α)+cos(180)sin(α)
=0.cos(α)+(-1)sin(α)
=-sin(α)
cos(180+α)
=cos(180)cos(α)-sin(180)sin(α)
=(-1)cos(α)-(0)sin(α)
=-cos(α)
for tan(180+α)
use (tan A + tan B)/(1 - (tan A)(tan B))
B.
Since both vectors are unit vectors, the cosine of the included angle is simply the dot product of A(cos(a),sin(a)) and B(cos(b),sin(b)):
cos(θ)
=(cos(a)cos(b)+sin(a)sin(b))/(|A| |B|)
=(cos(a-b))/(|1| |1|)
=cos(a-b)
therefore θ=a-b or -(a-b)
The magnitude of the cross product of the two vectors represent the positive area of a parallelogram formed by the two vectors as adjacent sides, namely,
Area=ABsin(θ)=|A x B|
|sin(θ)|
=|cos(a)sin(b)-cos(b)sin(a)|
=|sin(a-b)|
therefore
θ=|a-b|
Also, since the terminal points of the vectors A and B represent points on the unit circle at angles a and b from the x-axis, we conclude that the angle between the vectors is |a-b|.
A. To find simpler equivalent expressions, we can use the periodicity properties of the trigonometric functions.
(a) sin(180 + è) = sin(180)cos(è) + cos(180)sin(è)
= 0(cos(è)) + (-1)(sin(è))
= -sin(è)
(b) cos(180 + è) = cos(180)cos(è) - sin(180)sin(è)
= (-1)(cos(è)) - 0(sin(è))
= -cos(è)
(c) tan(180 + è) = sin(180 + è)/cos(180 + è)
= -sin(è)/(-cos(è))
= sin(è)/cos(è)
= tan(è)
In summary:
(a) sin(180 + è) = -sin(è)
(b) cos(180 + è) = -cos(è)
(c) tan(180 + è) = tan(è)
B. To calculate the angle formed by the vectors [cos 19, sin 19] and [cos 54, sin 54], we can use the dot product formula:
θ = arccos((u · v) / (|u| |v|))
where u and v are the given vectors, and |u| and |v| are the magnitudes of u and v, respectively.
First, let's calculate the dot product:
u · v = (cos 19)(cos 54) + (sin 19)(sin 54)
= cos(19 - 54) (using the identity cos(a - b) = cos a cos b + sin a sin b)
= cos(-35)
= cos 35
Next, let's calculate the magnitudes:
|u| = √((cos 19)^2 + (sin 19)^2)
= √(cos^2 19 + sin^2 19)
= √1 (using the identity cos^2 θ + sin^2 θ = 1)
= 1
|v| = √((cos 54)^2 + (sin 54)^2)
= √(cos^2 54 + sin^2 54)
= √1 (using the identity cos^2 θ + sin^2 θ = 1)
= 1
Now, substitute the values into the formula:
θ = arccos((cos 35) / (1 * 1))
= arccos(cos 35)
= 35 (since 0° ≤ θ ≤ 180°)
Therefore, the angle formed by the vectors [cos 19, sin 19] and [cos 54, sin 54] is 35 degrees.
A. To find simpler, equivalent expressions for the trigonometric functions with angles in the form of 180 + è, we can use the trigonometric identities that relate the functions with angles of the form 180 - è.
(a) sin(180 + è):
We know that sin(180 - è) = sin(è), based on the identity sin(180 - x) = sin(x). Using this identity, we can substitute 180 - è in place of è to get sin(180 + è) = sin(180 - (-è)) = sin(180 - (-è)) = sin(è). Therefore, sin(180 + è) is equivalent to sin(è).
(b) cos(180 + è):
Similarly, cos(180 - è) = -cos(è), based on the identity cos(180 - x) = -cos(x). Substituting 180 - è in place of è, we get cos(180 + è) = cos(180 - (-è)) = -cos(è). Hence, cos(180 + è) is equivalent to -cos(è).
(c) tan(180 + è):
Using the tangent identity, tan(x + 180) = tan(x), we can substitute 180 - è in place of x to get tan(180 + è) = tan(180 - (-è)) = tan(è). Therefore, tan(180 + è) is equivalent to tan(è).
B. To calculate the angle formed by the vectors [cos 19, sin 19] and [cos 54, sin 54], we can use the dot product formula and the arccosine function.
The dot product of two vectors [a, b] and [c, d] is given by a*c + b*d. In this case, the dot product of the given vectors is:
cos(19)*cos(54) + sin(19)*sin(54)
To calculate this, we need to convert the angles from degrees to radians. Since trigonometric functions usually work with radians, we'll use the radian measure.
To convert degrees to radians, use the conversion factor: π/180. So the angles become:
19 degrees = 19 * (π/180) radians
54 degrees = 54 * (π/180) radians
Now we can calculate the dot product using radians:
cos(19*(π/180)) * cos(54*(π/180)) + sin(19*(π/180)) * sin(54*(π/180))
Next, simplify the expression:
cos(19π/180) * cos(54π/180) + sin(19π/180) * sin(54π/180)
Evaluate the trigonometric functions using a calculator or any mathematical software. Once you have the result, use the arccosine function (usually denoted as acos) to find the angle:
angle = acos(dot product)
Repeat the process using the negative of the dot product: angle = acos(-dot product). This is because the arccosine function has a range between 0 and π, meaning only the acute angle can be obtained using this function. However, the angle formed by the vectors can also be reflex (greater than π), and taking the negative of the dot product allows us to calculate the reflex angle.
By performing these calculations using the dot product and the arccosine function with both the original dot product and its negation, you can find at least two ways to calculate the angle formed by the given vectors.