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nata0808 [166]
2 years ago
12

Use the triangle below to write the given ratios. • sin(P) = tan(T) = cos(T) =​

Mathematics
1 answer:
Sati [7]2 years ago
3 0

Answer:

.

Explaining:

The ratios of the sides of a right triangle are called trigonometric ratios. Three common trigonometric ratios are the sine (sin), cosine (cos), and tangent (tan). These are defined for acute angle.

In these definitions, the terms opposite, adjacent, and hypotenuse refer to the lengths of the sides.

You might be interested in
The graph of an exponential function is given. Which of the following is the correct equation of the function?
katen-ka-za [31]

Answer:

If one of the data points has the form  

(

0

,

a

)

, then a is the initial value. Using a, substitute the second point into the equation  

f

(

x

)

=

a

(

b

)

x

, and solve for b.

If neither of the data points have the form  

(

0

,

a

)

, substitute both points into two equations with the form  

f

(

x

)

=

a

(

b

)

x

. Solve the resulting system of two equations in two unknowns to find a and b.

Using the a and b found in the steps above, write the exponential function in the form  

f

(

x

)

=

a

(

b

)

x

.

EXAMPLE 3: WRITING AN EXPONENTIAL MODEL WHEN THE INITIAL VALUE IS KNOWN

In 2006, 80 deer were introduced into a wildlife refuge. By 2012, the population had grown to 180 deer. The population was growing exponentially. Write an algebraic function N(t) representing the population N of deer over time t.

SOLUTION

We let our independent variable t be the number of years after 2006. Thus, the information given in the problem can be written as input-output pairs: (0, 80) and (6, 180). Notice that by choosing our input variable to be measured as years after 2006, we have given ourselves the initial value for the function, a = 80. We can now substitute the second point into the equation  

N

(

t

)

=

80

b

t

to find b:

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎩

N

(

t

)

=

80

b

t

180

=

80

b

6

Substitute using point  

(

6

,

180

)

.

9

4

=

b

6

Divide and write in lowest terms

.

b

=

(

9

4

)

1

6

Isolate  

b

using properties of exponents

.

b

≈

1.1447

Round to 4 decimal places

.

NOTE: Unless otherwise stated, do not round any intermediate calculations. Then round the final answer to four places for the remainder of this section.

The exponential model for the population of deer is  

N

(

t

)

=

80

(

1.1447

)

t

. (Note that this exponential function models short-term growth. As the inputs gets large, the output will get increasingly larger, so much so that the model may not be useful in the long term.)

We can graph our model to observe the population growth of deer in the refuge over time. Notice that the graph below passes through the initial points given in the problem,  

(

0

,

8

0

)

and  

(

6

,

18

0

)

. We can also see that the domain for the function is  

[

0

,

∞

)

, and the range for the function is  

[

80

,

∞

)

.

Graph of the exponential function, N(t) = 80(1.1447)^t, with labeled points at (0, 80) and (6, 180).If one of the data points has the form  

(

0

,

a

)

, then a is the initial value. Using a, substitute the second point into the equation  

f

(

x

)

=

a

(

b

)

x

, and solve for b.

If neither of the data points have the form  

(

0

,

a

)

, substitute both points into two equations with the form  

f

(

x

)

=

a

(

b

)

x

. Solve the resulting system of two equations in two unknowns to find a and b.

Using the a and b found in the steps above, write the exponential function in the form  

f

(

x

)

=

a

(

b

)

x

.

EXAMPLE 3: WRITING AN EXPONENTIAL MODEL WHEN THE INITIAL VALUE IS KNOWN

In 2006, 80 deer were introduced into a wildlife refuge. By 2012, the population had grown to 180 deer. The population was growing exponentially. Write an algebraic function N(t) representing the population N of deer over time t.

SOLUTION

We let our independent variable t be the number of years after 2006. Thus, the information given in the problem can be written as input-output pairs: (0, 80) and (6, 180). Notice that by choosing our input variable to be measured as years after 2006, we have given ourselves the initial value for the function, a = 80. We can now substitute the second point into the equation  

N

(

t

)

=

80

b

t

to find b:

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎩

N

(

t

)

=

80

b

t

180

=

80

b

6

Substitute using point  

(

6

,

180

)

.

9

4

=

b

6

Divide and write in lowest terms

.

b

=

(

9

4

)

1

6

Isolate  

b

using properties of exponents

.

b

≈

1.1447

Round to 4 decimal places

.

NOTE: Unless otherwise stated, do not round any intermediate calculations. Then round the final answer to four places for the remainder of this section.

The exponential model for the population of deer is  

N

(

t

)

=

80

(

1.1447

)

t

. (Note that this exponential function models short-term growth. As the inputs gets large, the output will get increasingly larger, so much so that the model may not be useful in the long term.)

We can graph our model to observe the population growth of deer in the refuge over time. Notice that the graph below passes through the initial points given in the problem,  

(

0

,

8

0

)

and  

(

6

,

18

0

)

. We can also see that the domain for the function is  

[

0

,

∞

)

, and the range for the function is  

[

80

,

∞

)

.

Graph of the exponential function, N(t) = 80(1.1447)^t, with labeled points at (0, 80) and (6, 180).

Step-by-step explanation:

4 0
3 years ago
Consider the quadratic equation ax^2+bx+5=0,where a, b and c are rational numbers and the quadratic has two distinct zeros.
FrozenT [24]
You have shared the situation (problem), except for the directions:  What are you supposed to do here?  I can only make a educated guesses.  See below:

Note that if    <span>ax^2+bx+5=0    then it appears that c = 5 (a rational number).

Note that for simplicity's sake, we need to assume that the "two distinct zeros" are real numbers, not imaginary or complex numbers.  If this is the case, then the discriminant,    b^2 - 4(a)(c), must be positive.  Since c = 5, 

b^2 - 4(a)(5) > 0, or b^2 - 20a > 0.

Note that if the quadratic has two distinct zeros, which we'll call "d" and "e," then 

(x-d) and (x-e) are factors of ax^2 + bx + 5 = 0, and that because of this fact,

         - b plus sqrt( b^2 - 20a )
d =  ------------------------------------
                      2a

and

 </span>         - b minus sqrt( b^2 - 20a )
e =  ------------------------------------
                      2a

Some (or perhaps all) of these facts may help us find the values of "a" and "b."  Before going into that, however, I'm asking you to share the rest of the problem statement.  What, specificallyi, were you asked to do here?

7 0
3 years ago
Condense the expression to the logarithm of a single quantity.<br><br>​
vladimir1956 [14]

Answer:

\log _3\left(\frac{x^{\frac{1}{2}}}{\left(y+8\right)^2}\right)

Step-by-step explanation:

-> Apply log rules

\log _3\left(x^{\frac{1}{2}}\right)-2\log _3\left(y+8\right)

->

\log _3\left(x^{\frac{1}{2}}\right)-\log _3\left(\left(y+8\right)^2\right)

-> Apply basic math rules

\log _3\left(\frac{x^{\frac{1}{2}}}{\left(y+8\right)^2}\right)

7 0
2 years ago
A researcher uses an anonymous survey to investigate the television-viewing habits of American adolescents. Based on the set of
kow [346]

Answer:

The answer is - sample.

Step-by-step explanation:

Based on the set of 356 surveys that were completed and returned, the researcher finds that these students spend an average of 3.1 hours each day watching television.

For this study, the set of 356 students who returned the surveys is an example of a SAMPLE.

A sample is defined as a small part that shows what the whole population is like.

8 0
3 years ago
Do you multiply or add,subtract,divide to get answer:200 12.6 and 21.5
mrs_skeptik [129]
What is the answer you want to get?
3 0
3 years ago
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