Question

Sansa was explaining the meaning and usefulness of the statement tan 40° = 0.84. Which true statements below could be part of that explanation? Select all that apply.

a. All right triangles with an acute angle of 40° are similar to each other.
b. The sum of the squares of the legs of right triangles with a 40° angle equals 402.
c. Knowing that tan 40° = 0.84 is enough information to calculate the sides of any 40°-50°-90° triangle.
d. If you know tan 40° = 0.84 and the length of the leg opposite to the 40° angle, then you can calculate the length of the other leg.
e. The ratio of the opposite side to the hypotenuse in any right triangle with an angle of 40° is always approximately 0.84.

Answer 1

The statement tan 40° = 0.84 helps in understanding similar right triangles, calculating sides of a 40°-50°-90° triangle, and finding the lengths of legs given one side. The tangential value is a ratio of the opposite side to the adjacent side in right triangles.

Step-by-step explanation:

Understanding the statement tan 40° = 0.84 involves recognizing the relationships in right triangles and the basic principles of trigonometry. The value 0.84 represents the ratio of the opposite side to the adjacent side of a right triangle where one of the angles is 40°.

1. All right triangles with an acute angle of 40° are similar to each other. This is because they will all have the same angle ratios.
2. Knowing that tan 40° = 0.84 is enough information to calculate the sides of any 40°-50°-90° triangle, given one side. Using the ratio and the Pythagorean theorem, all sides can be determined.
3. If you know tan 40° = 0.84 and the length of the leg opposite to the 40° angle, then you can calculate the length of the adjacent leg. This is the practical use of the tangent function.

Answer choices 'a', 'c', and 'd' are all valid explanations of what it means when we say tan 40° = 0.84. However, choice 'b' is incorrect since it muddles the Pythagorean theorem, and 'e' incorrectly associates the tangent value with the ratio of the opposite side to the hypotenuse, when in fact, tangent relates to the opposite and adjacent sides.