Question

COMPLETION
30°
0%
Work out the value of x.
X
6 cm
3.5 cm

Answer 1

8.4 cm

Explanation:

We can find the altitude of the triangle by the Pythagorean theorem

et h be the altitude

`h = \sqrt{6^(2) - 3.5^(2)}\\= \sqrt{6^(2) - 3.5^(2)}\\\\= \sqrt{6^(2) - 3.5^(2)}\\\\= √(23.75)\\= 4.8734`

In the triangle on the right, notice it is a 30-60-90 right triangle. Such a triangle has the property that the side opposite 30°, ( h) is related t the side opposite 60° (x) by the relationship

x/sin 60 = a /sin 30

sin 60 = √3/2 and sin 30 = 1/2

so x = a/sin 30 x sin 60
x = [a/(1/2) ]x √3/2 = √3a

The vertical leg (4.8734) is related to the horizontal leg (x) by the formula

x = 4.8734 x √3

= 8.4 cm

Answer 2

Explanation:

first Pythagoras with the right, smaller right-angled triangle. that gives us the height of the total triangle, and the shorter leg of the left, larger right-angled triangle.

c² = a² + b²

c being the Hypotenuse (the side opposite of the 90° angle), a and b are the legs.

6² = 3.5² + height²

36 = 12.25 + height²

height² = 23.75

height = sqrt(23.75) = 4.873397172... cm

now, as mentioned, this height is also the second leg of the left, larger right-angled triangle.

and with a given angle, now trigonometry has to be used.

remember, what sine and cosine are ?

sine of an angle is the vertical (up/down) leg, and cosine is the horizontal (left/right) leg of the right-angled triangle created by the angle and inscribed in the (norm-)circle.

"norm-circle", because the radius is 1.

now, for a larger circumscribing circle (like in our case here), we need to multiply the actual radius by sine and cosine (and all other trigonometric functions).

the radius is the large, top-left side coming from the 30° angle.

we don't know the length yet, but we know the value of sin(30) multiplied by that length, which is the height we just calculated.

so,

sqrt(23.75) = sin(30)×radius

radius = sqrt(23.75)/sin(30)

sin(30) = 0.5 = 1/2

so,

radius = sqrt(23.75)/ 1/2 = sqrt(23.75)×2

x is now the cosine of the angle multiplied by the radius.

x = cos(30)×sqrt(23.75)×2 = 8.440971508... cm

≈ 8.4 cm