Question

Знайдіть синус і тангенс кута альфа, якщо косинус альфа 1/2 ?​

Answer 1

$\sin {}^{2} \alpha + \cos {}^{2} \alpha = 1 \to \: \sin \alpha = \sqrt{1 - \cos {}^{2} \alpha } = \pm \sqrt{1 - 0.5 {}^{2} } = \pm \sqrt{1 - 0.25} = \pm \sqrt{0.75} = \pm \sqrt{ \dfrac{3}{4} } = \pm \dfrac{ \sqrt{3} }{2} . \\ \\ \tg \alpha = \dfrac{ \sin \alpha }{ \cos \alpha } \to \: \dfrac{ \dfrac{ \sqrt{3} }{2} }{ \dfrac{1}{2} } = \dfrac{ \sqrt{3} }{1} = \sqrt{3} .$

Answer 2

Finding the Sine and Tangent of an Angle

To find the sine and tangent of an angle alpha, given that the cosine of alpha is 1/2, we can use trigonometric identities and formulas.

The cosine of an angle is defined as the ratio of the length of the adjacent side to the hypotenuse in a right triangle. In this case, we know that the cosine of alpha is 1/2. Let's denote the adjacent side as 'a', the opposite side as 'b', and the hypotenuse as 'c'.

Using the Pythagorean theorem, we can relate the lengths of the sides of a right triangle:

a^2 + b^2 = c^2

Since we know that the cosine of alpha is 1/2, we can substitute the values into the equation:

(1/2)^2 + b^2 = c^2

Simplifying, we get:

1/4 + b^2 = c^2

Now, let's solve for b^2:

b^2 = c^2 - 1/4

To find the sine of alpha, we can use the formula:

sin(alpha) = b/c

Substituting the values, we get:

sin(alpha) = sqrt(b^2) / c

sin(alpha) = sqrt(c^2 - 1/4) / c

Similarly, to find the tangent of alpha, we can use the formula:

tan(alpha) = b/a

Substituting the values, we get:

tan(alpha) = sqrt(c^2 - 1/4) / a

Now, let's calculate the values using the given information.

Sine of alpha: Using the equation sin(alpha) = sqrt(c^2 - 1/4) / c, we can substitute the value of c as 1 (since the cosine of alpha is 1/2):

sin(alpha) = sqrt(1^2 - 1/4) / 1

sin(alpha) = sqrt(3/4) / 1

sin(alpha) = sqrt(3) / 2

Therefore, the sine of alpha is sqrt(3) / 2.

Tangent of alpha: Using the equation tan(alpha) = sqrt(c^2 - 1/4) / a, we can substitute the value of c as 1 (since the cosine of alpha is 1/2) and a as 1 (since a is the adjacent side):

tan(alpha) = sqrt(1^2 - 1/4) / 1

tan(alpha) = sqrt(3/4) / 1

tan(alpha) = sqrt(3) / 2

Therefore, the tangent of alpha is sqrt(3) / 2.

Please note that the values of the sine and tangent of alpha are both equal to sqrt(3) / 2.