If A is acute and tan A = x/1, determine the values of the remaining
functions. Provide an Illustration.
a = x
b = 1
c = √(1+x^2)
sinA = a/c
and so on
Well, if A is acute and tan A = x/1, we can use the Pythagorean identity to find the other trigonometric functions. The Pythagorean identity states that sin^2(A) + cos^2(A) = 1.
Since tan A = x/1, we can let the opposite side be x and the adjacent side be 1. Using the Pythagorean theorem, we can find the hypotenuse: hypotenuse^2 = opposite^2 + adjacent^2. In this case, hypotenuse^2 = x^2 + 1^2.
Now, we can find the values of sin A and cos A using the sides of our right triangle. sin A = opposite/hypotenuse = x / √(x^2 + 1^2), and cos A = adjacent/hypotenuse = 1 / √(x^2 + 1^2).
Here’s an illustration of the right triangle:
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x | 1
____________|____________
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|
I hope this helps! If not, I'll be here all night, so I can tell you some more terrible jokes to make up for it.
To determine the values of the remaining trigonometric functions, we first need to find the values of the sides of the triangle.
Given that tan A = x/1, we can interpret this as the ratio of the length of the side opposite angle A (which we'll call x) to the length of the adjacent side (which we'll call 1).
Using the Pythagorean theorem (a^2 + b^2 = c^2), we can find the length of the hypotenuse (c).
In this case, a = x, b = 1, and c is the hypotenuse.
Using the formula, we obtain:
(x)^2 + (1)^2 = c^2
x^2 + 1 = c^2
Next, we can find the values of the remaining trigonometric functions using the definitions:
sin A = opposite/hypotenuse
cos A = adjacent/hypotenuse
cosec A = 1/sin A (reciprocal of sine)
sec A = 1/cos A (reciprocal of cosine)
cot A = 1/tan A (reciprocal of tangent)
To illustrate this, we can draw a right triangle with angle A, where the length of the side opposite A is x and the length of the adjacent side is 1. The length of the hypotenuse can be found by solving the equation x^2 + 1 = c^2.
Here is an illustration:
```
/|
/ |
/ | x
/ |
/ A |
1 /____|
```
In this triangle, angle A is acute, and the side opposite A has a length of x, and the adjacent side has a length of 1. The length of the hypotenuse, represented by c, can be found by solving the equation x^2 + 1 = c^2.
Once we have the value of c, we can calculate the values of the remaining functions using the definitions mentioned earlier.
Note that without a specific value of x given, we cannot provide the exact values of the trigonometric functions. However, the steps described above will help you determine the values once you know the value of x.
To determine the values of the remaining trigonometric functions (sine, cosine, secant, cosecant, and cotangent) in terms of variable x, we can use the given information that tan A = x/1.
First, recall that the tangent function is defined as the ratio of the opposite side (in this case, x) to the adjacent side (in this case, 1) in a right triangle. Since we know A is an acute angle, we can draw a right triangle and label one of the acute angles as A.
Let's draw a right triangle and label the sides:
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| \
x | \ 1
| \
|______\
In this triangle, the angle A is at the top vertex. The side opposite angle A is x, and the side adjacent to angle A is 1.
Now, we can use the Pythagorean theorem to find the length of the hypotenuse. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. In this case, we have:
hypotenuse^2 = x^2 + 1^2
Let's solve for the hypotenuse:
hypotenuse^2 = x^2 + 1
hypotenuse = √(x^2 + 1)
Now that we know the length of the hypotenuse, we can determine the values of the remaining trigonometric functions:
1. Sine (sin A): The sine function is defined as the ratio of the opposite side (x) to the hypotenuse (√(x^2 + 1)). Therefore, sin A = x/√(x^2 + 1).
2. Cosine (cos A): The cosine function is defined as the ratio of the adjacent side (1) to the hypotenuse (√(x^2 + 1)). Therefore, cos A = 1/√(x^2 + 1).
3. Secant (sec A): The secant function is the reciprocal of the cosine function. Therefore, sec A = 1/cos A = √(x^2 + 1)/1.
4. Cosecant (csc A): The cosecant function is the reciprocal of the sine function. Therefore, csc A = 1/sin A = √(x^2 + 1)/x.
5. Cotangent (cot A): The cotangent function is the reciprocal of the tangent function. Therefore, cot A = 1/tan A = 1/(x/1) = 1/x.
In summary, the values of the remaining trigonometric functions are as follows:
sin A = x/√(x^2 + 1)
cos A = 1/√(x^2 + 1)
sec A = √(x^2 + 1)/1
csc A = √(x^2 + 1)/x
cot A = 1/x
Please note that the values of these functions will vary depending on the value of x.