Let's go birdwatching together, and see the various birds in nature hunt down their prey using trigonometry!
The tangent function will give an output that is equal to the opposite over the adjacent. So in order to find the angle θ, using the tan-1 we need to plug in the opposite over adjacent (10/7) and we get angle of ≈ 55°.
Let's calculate the angle this bird has to fly on to get to the worm, using tan-1.
We use sin as it relates the side opposite and hypotenuse, which are the one (two if you include theta) know and unknown we have and are looking for. Plugging in numbers we get that the sin(70) = 24m/c. If we multiply by c and divided by sin(70) we get a rearranged equation that tells us c = 24/sin(70). This gives us that length of c is ≈ 25.54m.
Here, all we care about is how far away the bird is from the nut, we don't need to find out the length of the leg. Using the sin function we can do that find the distance.
Finally, we have our last bird, we once again want to find the displacement to the fish so we use cos.
Here, once again we use cos as it relates the side adjacent to hypotenuse. We get that cos(21) = 7m/c, so flipping around to rearrange the variable we get c = 7/cos(21). And therefore the length of c is ≈ 7.5m.