How do i convince my parents to let me get a phone?
2 answers:
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Answer:
75.7°
Step-by-step explanation:
The mnemonic SOH CAH TOA is intended to remind you of the relations between trig functions and sides of a right triangle. You are given all three sides of the triangle, so you can make use of at least two different trig functions to find the missing angle.
Cos = Adjacent/Hypotenuse
Tan = Opposite/Adjacent
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<h3>cosine</h3>The hypotenuse is 65, and the side adjacent to the unknown angle is 16. That tells you ...
cos(?) = 16/65
The inverse function is used to find the angle value:
? = arccos(16/65) ≈ 75.7°
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<h3>tangent</h3>The side opposite the angle of interest is 63. Then you have ...
tan(?) = 63/16
The inverse function is used to find the angle value:
? = arctan(63/16) ≈ 75.7°
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<em>Additional comments</em>
When using trig functions on a calculator, you need to make sure the angle mode is set to what you want. Here, we want angles in degrees, so we have set that as the angle mode. The [DEG] icon in the lower left corner of the display confirms this.
We can't tell what you're supposed to round the value to. The attachment gives enough digits for you to be able to round to whatever precision you need.
Answer:
Proofs contained within the explanation.
Step-by-step explanation:
These induction proofs will consist of a base case, assumption of the equation holding for a certain unknown natural number, and then proving it is true for the next natural number.
a)
Proof
Base case:
We want to shown the given equation is true for n=1:
The first term on left is 2 so when n=1 the sum of the left is 2.
Now what do we get for the right when n=1:
So the equation holds for n=1 since this leads to the true equation 2=2:
We are going to assume the following equation holds for some integer k greater than or equal to 1:
Given this assumption we want to show the following:
Let's start with the left hand side:
The first k terms we know have a sum of .5k(3k+1) by our assumption.
Distribute for the second term:
Combine terms in second term:
Factor out a half from both terms:
Distribute for both first and second term in the [ ].
Combine like terms in the [ ].
The thing inside the [ ] is called a quadratic expression. It has a coefficient of 3 so we need to find two numbers that multiply to be ac (3*4) and add up to be b (7).
Those numbers would be 3 and 4 since
3(4)=12 and 3+4=7.
So we are going to factor by grouping now after substituting 7k for 3k+4k:
.
Therefore for all integers n equal or greater than 1 the following equation holds:
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b)
Proof:
Base case: When n=1, the left hand side is 1.
The right hand at n=1 gives us:
So both sides are 1 for n=1, therefore the equation holds for the base case, n=1.
We want to assume the following equation holds for some natural k:
.
We are going to use this assumption to show the following:
Let's start with the left side:
We know the sum of the first k terms is 1/4(5^k-1) given by our assumption:
Factor out the 1/4 from both of the two terms:
Combine the like terms inside the [ ]:
Apply law of exponents:
Therefore the following equation holds for all natural n:
.
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