Question

extra credit
find the value of each trigonometric ratio.

1. tan x
2. cos c
3. sin a
4. cos a
5. cos c
6. sin x
7. tan a
8. sin a

find the missing side. round to the nearest tenth.
13)
14)
15)
16)

Explanation:

Let's solve problem 5) first: finding \(\tan X\) in the right triangle with legs \(XY = 12\) and \(YZ = 35\), and hypotenuse \(XZ = 37\).

Step 1: Recall the definition of tangent

In a right triangle, the tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side. For \(\angle X\), the opposite side to \(\angle X\) is \(YZ = 35\) and the adjacent side is \(XY = 12\). So, \(\tan X=\frac{\text{opposite}}{\text{adjacent}}=\frac{YZ}{XY}\).

Step 2: Substitute the values

We know \(YZ = 35\) and \(XY = 12\), so \(\tan X=\frac{35}{12}\approx2.9167\) (if we want a decimal approximation) or we can leave it as a fraction \(\frac{35}{12}\). But usually, for such problems, we simplify the fraction or give the decimal. Since \(35\div12 = 2\frac{11}{12}\approx2.92\) (rounded to two decimal places) or as a reduced fraction \(\frac{35}{12}\). But let's check the triangle again. Wait, the right angle is at \(Y\), so the sides: \(XY = 12\) (one leg), \(YZ = 35\) (the other leg), and \(XZ = 37\) (hypotenuse). So for \(\angle X\), the opposite side is \(YZ = 35\), adjacent is \(XY = 12\). So \(\tan X=\frac{35}{12}\).

Answer:

\(\frac{35}{12}\) (or approximately \(2.92\))