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3 years ago

In the triangle ABC, if side a is 3, side b is 4 and side c is 5, what is the cosine of the smallest angle?

Mathematics

1 answer:

sashaice [31]3 years ago

5 0

Answer:

Cosine of the smallest angle is 4/5.

Step-by-step explanation:

It is given that in the triangle ABC, side a is 3, side b is 4 and side c is 5.

Sum of squares of two smaller sides.

$3^2+4^2=9+16=25$

Sum of squares of largest sides.

$5^2=25$

Since sum of squares of two smaller sides is equal to sum of squares of largest sides, therefore triangle ABC is a right angle triangle.

Hypotenuse = 5 units.

In a right angle triangle, the smallest angle has shortest opposite side.

Shortest side is a=3 It means angle A is smallest.

$\cos \theta = \dfrac{adjacent}{hypotenuse}$

$\cos (A) = \dfrac{AC}{AB}$

$\cos (A) = \dfrac{4}{5}$

Therefore, the cosine of the smallest angle is 4/5.

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2 years ago

If r and s are positive integers, is \small \frac{r}{s} an integer? (1) Every factor of s is also a factor of r. (2) Every prime

Yuri [45]

Answer:

If statement(1) holds true, it is correct that $\small \frac{r}{s}$ is an integer.

If statement(2) holds true, it is not necessarily correct that $\small \frac{r}{s}$ is an integer.

Step-by-step explanation:

Given two positive integers $r$ and $s$ .

To check whether $\small \frac{r}{s}$ is an integer:

Condition (1):

Every factor of $s$ is also a factor of $r$ .

$r \geq s$

Let us consider an example:

$s = 5^2 \cdot 2\\r = 5^3 \cdot 2^2$

$\dfrac{r}{s} = \dfrac{5^3\cdot2^2}{5^2\cdot2} = 10$

which is an integer.

Actually, in this situation $s$ is a factor of $r$ .

Condition 2:

Every prime factor of s is also a prime factor of r.

(But the powers of prime factors need not be equal as we are not given the conditions related to powers of prime factors.)

Let

$r = 2^2\cdot 5\\s =2^4\cdot 5$

$\dfrac{r}{s} = \dfrac{2^3\cdot5}{2^4\cdot5} = \dfrac{1}{2}$

which is not an integer.

So, the answer is:

If statement(1) holds true, it is correct that $\small \frac{r}{s}$ is an integer.

If statement(2) holds true, it is not necessarily correct that $\small \frac{r}{s}$ is an integer.

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3 years ago

If Log 20 = 1.301 what is the value of Logsub100 20?

Charra [1.4K]

Answer:

$log_{100}20 = 0.6505$

Step-by-step explanation:

log20 = 1.301 (when you see only "log" with no base, it is taken as a base 10)

This basically means:

$10^{1.301} = 20$ (This is the log in exponential form)

Since we don't know what $log_{100}20$ is equal to, we will say $log_{100}20$ = x

So to solve for $log_{100}20$ = x, you do the same thing. (convert to exponential form)

$100^{x} =20$

Now you will notice that both of these equations are equal to 20.

Since 20 = 20,

we can say $100^{x}$ = $10^{1.301}$

Another way of saying 100, is $10^{2}$ (make the bases the same)

Now we get $10^{2x} = 10^{1.301}$

(we get 2x because an exponent to the power of an exponent ( $2^{x}$ ) is the same as 2 * x)

Because you have the same base, you can just ignore the 10s and focus on the exponents. So you get:

2x = 1.301

x = 0.6505

6 0

2 years ago