Find the value of sin 37°, sin 53°, tan 37°, tan 53° in terms of the fraction.
We can use complementary relations to find the values.
Answer: sin 37° = 3/5, sin 53° = 4/5, tan 37° = 3/4, tan 53° = 4/3
Let's proceed step by step.
Explanation:
Note that 37° + 53° = 90°. Thus, we can construct a right triangle with angles 90°, 37°, and 53°.
Remember that a right triangle with these 3 angles 90°, 37°, and 53° is always associated with the Pythagorean triplet 3,4,5.
But how to identify which of these numbers go to the opposite of which of these angles? Recall that the size of an angle in a triangle is always proportional to the size of its opposite side. i.e., if an angle of a triangle is the biggest among all three angles, then its opposite side is the biggest of all three sides. Using this logic, we can arrange the above 3 angles and 3 sides as a triangle below:
Let's analyze the figure given below.
Here, 3 and 4 are the opposite and adjacent sides of the angle 37°; and 4 and 3 are the opposite and adjacent sides of the angle 53° respectively.
Now, by just applying the definitions of trigonometric ratios:
- sin A = opposite / hypotenuse ⇒ sin 37° = 3/5 and sin 53° = 4/5
- tan A = opposite / adjacent ⇒ tan 37° = 3/4 and tan 53° = 4/3