In a right triangle, angle A is 26°. If that angle is increased, at which angle measure will the value of tan A be double the value of tan 26°?
42.8°
47.5°
44.3°
45.4°
Let's find the value of tan 26° to determine the angle measure that would result in double the value.
We know that tan 26° = opposite/adjacent.
By drawing the right triangle and labeling the opposite side as O and the adjacent side as A, we can use the given information to calculate the value of tan 26°.
tan 26° = O/A.
To find O and A, let's assume that the hypotenuse is equal to 1.
Since we know the value of one angle (26°) in a right triangle, we can use trigonometric ratios to find the lengths of the sides. We can use the following ratios:
sin A = opposite/hypotenuse
cos A = adjacent/hypotenuse
tan A = opposite/adjacent
In this case, we want to find tan 26°, so we rearrange the formula to solve for O/A:
tan A = O/A
O/A = tan A
Since hypotenuse = 1, we know:
cos A = A (opposite side)
sin A = O (adjacent side)
Now we can rewrite the equation:
tan A = O/A
tan A = sin A/cos A
Using the values of sin A and cos A, we can find tan A:
tan 26° = sin 26°/cos 26°
sin 26° = 0.43837 (rounded to five decimal places)
cos 26° = 0.89879 (rounded to five decimal places)
tan 26° = 0.43837/0.89879
tan 26° = 0.48773 (rounded to five decimal places)
We want to find the angle measure that would result in double the value of tan 26°.
2 * tan 26° = 2 * 0.48773
2 * tan 26° = 0.97546
To find the angle measure that has a tan value of 0.97546, we can use the inverse tan function.
Let x be the angle measure we are looking for.
tan x = 0.97546
x = tan^(-1)(0.97546)
x ≈ 44.37°
Since we are looking for the angle measure when it's increased, we need to round up:
x ≈ 45.4°
Therefore, the angle measure that would result in double the value of tan 26° is approximately 45.4°.
So the correct answer is 45.4°.