In ABC, vertex C is a right angle. Which trigonometric ratio has the same
trigonometric value as Sin A?
A Sin B
B Cosine A
C Cosine B
D Tan A
sin A = cos B
Cosine B
To determine which trigonometric ratio has the same value as sin A, we need to consider the angle A.
If vertex C is a right angle in triangle ABC, we can use the following trigonometric ratios to determine which one is equivalent to sin A:
- Sin A: This is the ratio of the length of the side opposite angle A to the length of the hypotenuse.
- Cosine A: This is the ratio of the length of the side adjacent to angle A to the length of the hypotenuse.
- Sin B: This is the ratio of the length of the side opposite angle B to the length of the hypotenuse.
- Cosine B: This is the ratio of the length of the side adjacent to angle B to the length of the hypotenuse.
- Tan A: This is the ratio of the length of the side opposite angle A to the length of the side adjacent to angle A.
Since we are looking for the trigonometric ratio with the same value as sin A, the correct answer is:
A) Sin B
To determine which trigonometric ratio has the same value as sin A, we first need to understand the relationship between the angles and trigonometric functions in a right triangle.
In a right triangle, the sine function is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse. So, sin A would be equal to the length of the side opposite angle A divided by the length of the hypotenuse.
Now, let's analyze the given options:
A) Sin B: When comparing sin A to sin B, we can see that angle B is not mentioned in the question. Therefore, we cannot determine if sin A and sin B have the same value. This option is not the correct choice.
B) Cosine A: The cosine function is defined as the ratio of the length of the adjacent side to the length of the hypotenuse. As sin A is the opposite side and cosine A is the adjacent side, they do not have the same value. This option is not the correct choice.
C) Cosine B: Just like option B, we do not have information about angle B, so we cannot determine the relationship between sin A and cosine B. This option is not the correct choice.
D) Tan A: The tangent function is defined as the ratio of the length of the side opposite to the angle to the length of the side adjacent to the angle. Since sin A is the ratio of the opposite side to the hypotenuse and tan A is the ratio of the opposite side to the adjacent side, they can have the same value. Therefore, the correct choice is option D, Tan A.
To summarize, the trigonometric ratio that has the same trigonometric value as sin A is Tan A (option D).