1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
True [87]
3 years ago
8

Prove it please answer only if you know​

Mathematics
1 answer:
deff fn [24]3 years ago
4 0

Part (c)

We'll use this identity

\sin(x+y) = \sin(x)\cos(y) + \cos(x)\sin(y)\\\\

to say

\sin(A+45) = \sin(A)\cos(45) + \cos(A)\sin(45)\\\\\sin(A+45) = \sin(A)\frac{\sqrt{2}}{2} + \cos(A)\frac{\sqrt{2}}{2}\\\\\sin(A+45) = \frac{\sqrt{2}}{2}(\sin(A)+\cos(A))\\\\

Similarly,

\sin(A-45) = \sin(A + (-45))\\\\\sin(A-45) = \sin(A)\cos(-45) + \cos(A)\sin(-45)\\\\\sin(A-45) = \sin(A)\cos(45) - \cos(A)\sin(45)\\\\\sin(A-45) = \sin(A)\frac{\sqrt{2}}{2} - \cos(A)\frac{\sqrt{2}}{2}\\\\\sin(A-45) = \frac{\sqrt{2}}{2}(\sin(A)-\cos(A))\\\\

-------------------------

The key takeaways here are that

\sin(A+45) = \frac{\sqrt{2}}{2}(\sin(A)+\cos(A))\\\\\sin(A-45) = \frac{\sqrt{2}}{2}(\sin(A)-\cos(A))\\\\

Therefore,

2\sin(A+45)*\sin(A-45) = 2*\frac{\sqrt{2}}{2}(\sin(A)+\cos(A))*\frac{\sqrt{2}}{2}(\sin(A)-\cos(A))\\\\2\sin(A+45)*\sin(A-45) = 2*\left(\frac{\sqrt{2}}{2}\right)^2\left(\sin^2(A)-\cos^2(A)\right)\\\\2\sin(A+45)*\sin(A-45) = 2*\frac{2}{4}\left(\sin^2(A)-\cos^2(A)\right)\\\\2\sin(A+45)*\sin(A-45) = \sin^2(A)-\cos^2(A)\\\\

The identity is confirmed.

==========================================================

Part (d)

\sin(x+y) = \sin(x)\cos(y) + \cos(x)\sin(y)\\\\\sin(45+A) = \sin(45)\cos(A) + \cos(45)\sin(A)\\\\\sin(45+A) = \frac{\sqrt{2}}{2}\cos(A) + \frac{\sqrt{2}}{2}\sin(A)\\\\\sin(45+A) = \frac{\sqrt{2}}{2}(\cos(A)+\sin(A))\\\\

Similarly,

\sin(45-A) = \sin(45 + (-A))\\\\\sin(45-A) = \sin(45)\cos(-A) + \cos(45)\sin(-A)\\\\\sin(45-A) = \sin(45)\cos(A) - \cos(45)\sin(A)\\\\\sin(45-A) = \frac{\sqrt{2}}{2}\cos(A) - \frac{\sqrt{2}}{2}\sin(A)\\\\\sin(45-A) = \frac{\sqrt{2}}{2}(\cos(A)-\sin(A))\\\\

-----------------

We'll square each equation

\sin(45+A) = \frac{\sqrt{2}}{2}(\cos(A)+\sin(A))\\\\\sin^2(45+A) = \left(\frac{\sqrt{2}}{2}(\cos(A)+\sin(A))\right)^2\\\\\sin^2(45+A) = \frac{1}{2}\left(\cos^2(A)+2\sin(A)\cos(A)+\sin^2(A)\right)\\\\\sin^2(45+A) = \frac{1}{2}\cos^2(A)+\frac{1}{2}*2\sin(A)\cos(A)+\frac{1}{2}\sin^2(A)\right)\\\\\sin^2(45+A) = \frac{1}{2}\cos^2(A)+\sin(A)\cos(A)+\frac{1}{2}\sin^2(A)\right)\\\\

and

\sin(45-A) = \frac{\sqrt{2}}{2}(\cos(A)-\sin(A))\\\\\sin^2(45-A) = \left(\frac{\sqrt{2}}{2}(\cos(A)-\sin(A))\right)^2\\\\\sin^2(45-A) = \frac{1}{2}\left(\cos^2(A)-2\sin(A)\cos(A)+\sin^2(A)\right)\\\\\sin^2(45-A) = \frac{1}{2}\cos^2(A)-\frac{1}{2}*2\sin(A)\cos(A)+\frac{1}{2}\sin^2(A)\right)\\\\\sin^2(45-A) = \frac{1}{2}\cos^2(A)-\sin(A)\cos(A)+\frac{1}{2}\sin^2(A)\right)\\\\

--------------------

Let's compare the results we got.

\sin^2(45+A) = \frac{1}{2}\cos^2(A)+\sin(A)\cos(A)+\frac{1}{2}\sin^2(A)\right)\\\\\sin^2(45-A) = \frac{1}{2}\cos^2(A)-\sin(A)\cos(A)+\frac{1}{2}\sin^2(A)\right)\\\\

Now if we add the terms straight down, we end up with \sin^2(45+A)+\sin^2(45-A) on the left side

As for the right side, the sin(A)cos(A) terms cancel out since they add to 0.

Also note how \frac{1}{2}\cos^2(A)+\frac{1}{2}\cos^2(A) = \cos^2(A) and similarly for the sin^2 terms as well.

The right hand side becomes \cos^2(A)+\sin^2(A) but that's always equal to 1 (pythagorean trig identity)

This confirms that \sin^2(45+A)+\sin^2(45-A) = 1 is an identity

You might be interested in
Be-1 where on the boat are registration numbers placed?
Neko [114]
<span>There is a requirement for the Coast Guard that all vessels must be registered with the state in which they'll be operated and their registration numbers must be clearly displayed on the boat.
</span><span>The sequence of the numbers must be written </span>to both sides of the vessel<span> in a sequence such that it can be read from left to right. The mirror-reading on the vessel is not allowed.
</span>Now the numbers must be Displayed on the front portion of the boat that is the forward half of the vessel.
6 0
3 years ago
What are the x- and y-intercepts of the graph of 3x+6y=12?
Dafna1 [17]

Answer:

X-int.: (4,0)

Y-int.: (0,2)

Step-by-step explanation:

3 0
3 years ago
Write the verbal sentence as an equation.<br> Fourteen minus the product of 3 and a number is 26.
Firdavs [7]

Answer:

14-3x=26

Step-by-step explanation:

When it says "is" and then a number, that means that it is the answer. In this case, 26.

Product means multiplying.

8 0
3 years ago
How can estimating be helpful
nadezda [96]
It helps you narrow down to a smaller answer for example there is pi, (3.14159265359) and there is more and more to that number. It is now estimated to 3.14 because it would be almost impossible to work with pi if it wasn't estimated
6 0
3 years ago
What is the GCF of, 33,22? I need this now.
Solnce55 [7]

Answer:

11

Step-by-step explanation:

33÷11=3

22÷11=2

3 and 2 habe no common factors

3 0
3 years ago
Other questions:
  • A circle has a diameter of 26 units. What is the area of the circle to the nearest hundredth of a square unit?
    6·2 answers
  • Solve the following :- 2n &gt; 14. n &gt; _____ . 2. n/5≥ 11. n ≥ _______. 3. –3y ≤ –18. y ≥ _______
    7·1 answer
  • 8 divided by 1766? please help me out
    13·2 answers
  • Seth’s parents gave him $5000 to invest for his 16th birthday. He is considering two investment options. Option A will pay him 4
    15·1 answer
  • Y = 4x – 10
    7·2 answers
  • Are you looking for surface are or volume?
    13·2 answers
  • 16. What happens to the equation when y = x becomes y = x + 3? 915<br> 291051 wood to
    11·1 answer
  • Classify the triangles shown below as "scalene," "isosceles," or "equilateral." Sides that are the same length are marked with a
    10·1 answer
  • Would a triangle with the sides of 16 m, 30 m, and 34 m make a right triangle?
    7·1 answer
  • Solve simultaneous equations <br> 11s+9t=116<br> 5s+9t=92
    6·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!