The trigonometric method of vector resolution involves using
trigonometric functions to determine the components of the vector. Earlier in
lesson 1, the use of trigonometric functions to
determine the direction of a vector was described. Now in this part of lesson
1, trigonometric functions will be used to determine the components of a single
vector. Recall from the earlier discussion that
trigonometric functions relate the ratio of the lengths of the sides of a right
triangle to the measure of an acute angle within the right triangle. As such,
trigonometric functions can be used to determine the length of the sides of a
right triangle if an angle measure and the length of one side are known.
The method of employing trigonometric functions to determine
the components of a vector are as follows:
1. Construct
a rough sketch (no scale needed) of the vector
in the indicated direction. Label its magnitude and the angle that it makes
with the horizontal.
2. Draw a
rectangle about the vector such that the vector is the diagonal of the
rectangle. Beginning at the tail of the vector, sketch
vertical and horizontal lines. Then sketch horizontal and vertical lines at the head of the
vector. The sketched lines will meet to form a rectangle.
3. Draw the
components of the vector. The components are the sides of the rectangle. The tail of each
component begins at the tail of the vector and stretches along the axes to the
nearest corner of the rectangle. Be sure to place arrowheads on these
components to indicate their direction (up, down, left, right).
4. Meaningfully
label the components of the vectors with symbols to indicate which component
represents which side. A northward force component might be labeled Fnorth. A
rightward force velocity component might be labeled vx; etc.
5. To
determine the length of the side opposite the indicated angle, use the sine
function. Substitute the magnitude of the vector for the length of the
hypotenuse. Use some algebra to solve the equation for the length of the side
opposite the indicated angle.
6. Repeat the
above step using the cosine function to determine the length of the side
adjacent to the indicated angle.
The above method is illustrated below for determining the
components of the force acting upon Fido. As the 60-Newton tension force acts
upward and rightward on Fido at an angle of 40 degrees, the components of this
force can be determined using trigonometric functions.