Mathematics, 16.07.2019 00:30 Carlyalexis77301

How do you find the sine, cosine, and tangent values on the unit circle? provide an example.

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In the unit circle sine and cosine are given by the y-coordinate and x-coordinate, respectively. Tangent is the ratio y-coordinate / x-coordinate.

That is so in virtue of the unit circle has radius 1. That implies that the hypotenuse is 1, so sine is opposite-leg / hypotenuse = y-ccordinate / 1; and cosine is adjacent-leg / hypotenuse = x-coordinate / 1.

Some examples are:

1) Angle 0 ⇒ x-coordinate = 1, y-coordinate ⇒

cosine 0 = 1;
sine 0 = 0;
tangent = 0 / 1 = 0

2) Angle 45° ⇒ x-coordinate = (√2 ) / 2; y-coordinate = (√2) / 2 ⇒

cosine 45° = (√2 / 2)
sine 45° = (√2 ) / 2

tan 45° = 1

3) Angle 90° ⇒ x-coordinate = 0; y-coordinate = 1 ⇒

cos 90° = 0
sin 90° = 1
tan 90° = 1 / 0 = ∞ (undefined)

Answer from: yhhh

To find the sine, cosine, and tangent values on the unit circle:

cos a = coordinate value xsin a = coordinate value of ytan a = coordinate value y : coordinate value xFurther explanation

In trigonometry, there are 3 angular functions, namely sin, cos and tan, based on a right triangle

There are 3 sides used in this triangle, namely:

1. hypotenuse: the longest side2. opposite to the angle θ3. adjacent to the angle θ

General formulas for sin, cos, and tan:

$\displaystyle sin\:a=\frac{opposite}{hypotenuse}$

$\displaystyle cos\:a=\frac{adjacent}{hypotenuse}$

$\displaystyle tan\:a=\frac{opposite}{adjacent}$

There are 3 other functions:

$\displaystyle sec\:a=\frac{1}{cos\:a}$

$\displaystyle csc\:a=\frac{1}{sin\:a}$

$\displaystyle cot\:a=\frac{1}{tan\:a}$

The unit circle is a circle with a radius = 1

Because of the value of r = 1, the value :

cos a = coordinate value xsin a = coordinate value of ytan a = coordinate value y : coordinate value x

So the coordinate point in the circle is a point (cos a, sin a)

A Pythagorean theorem which states that:

the hypotenuse or the longest side in a right triangle equal to the sum of the squares of the other sides.

Then in the unit circle, the phytagorean equation becomes :

$\displaystyle x^2+y^2=1^2$

$\displaystyle x^2+y^2=1$

Example

the angle θ = 0 °, x = 1, y = 0

$\displaystyle sin\:a=\frac{opposite}{hypotenuse}=\frac{0}{1}=0$

$\displaystyle cos\:a=\frac{adjacent}{hypotenuse}=\frac{1}{1}=1$

$\displaystyle tan\:a=\frac{opposite}{adjacent}=\frac{0}{1}=0$

the angle θ = 90, x = 0, y = 1

$\displaystyle sin\:a=\frac{opposite}{hypotenuse}=\frac{1}{1}=1$

$\displaystyle cos\:a=\frac{adjacent}{hypotenuse}=\frac{0}{1}=0$

$\displaystyle tan\:a=\frac{opposite}{adjacent}=\frac{1}{0}=undfned$

Learn more

Pythagoras theorem

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trigonometric ratio to find x

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trigonometric identities

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Keywords: unit circle, coordinate point, sin, cos, tan

How do you find the sine, cosine, and tangent values on the unit circle? provide an example.

Answer from: Quest

the solution of the equation is 2.23 ⇒ answer b

step-by-step explanation:

* lets revise the meaning of exponential function

- the form of the exponential function is y = ab^x, where a ≠ 0, b > 0 ,

b ≠ 1, and x is any real number

- it has a constant base b

- it has a variable exponent x

- to solve an exponential equation, take the log or ln of both sides,

and solve for the variable

* lets solve the problem

∵ $7^{x}=77$

- the base is 7 and the exponent is x

- insert ㏑ in both sides

∴ $ln(7^{x})=ln(77)$

- use the rule $ln(a^{n})= nln(a)$

∴ x ㏑(7) = ㏑(77)

- to find x divide both sides by ㏑(7)

∴ $x=\frac{ln(77)}{ln(7)}=2.23$

* the solution of the equation is 2.23 to the nearest 2 decimal places

Answer from: Quest

step-by-step explanation:

659

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