A right triangle has two acute angles, and theta; phi so that cos phi = x + 0.2 and sin theta = 2x - 0.5 . What is the value of x? ( 1 point) 0.07 10 0.7 1
To solve this problem, we need to use the trigonometric identities for cosine and sine.
The value of cosine is defined as the ratio of the adjacent side to the hypotenuse in a right triangle. So, we can write the equation: cos(phi) = adjacent/hypotenuse.
Given that cos(phi) = x + 0.2, we can rewrite the equation as follows: x + 0.2 = adjacent/hypotenuse.
Similarly, the value of sine is defined as the ratio of the opposite side to the hypotenuse in a right triangle. So, we can write the equation: sin(theta) = opposite/hypotenuse.
Given that sin(theta) = 2x - 0.5, we can rewrite the equation as follows: 2x - 0.5 = opposite/hypotenuse.
Since two acute angles are involved, we can consider two right triangles that share the same hypotenuse.
By comparing the equations, we can see that the hypotenuse remains the same in both equations.
Let's multiply both equations by the hypotenuse to eliminate the fraction:
x + 0.2 = adjacent,
2x - 0.5 = opposite.
Since the two right triangles share the same hypotenuse, the adjacent side in one triangle is the opposite side in the other triangle.
So, we can set adjacent = opposite and solve the system of equations:
x + 0.2 = 2x - 0.5.
Rearranging the equation, we have:
2x - x = 0.2 + 0.5,
x = 0.7.
Therefore, the value of x is 0.7.