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Section 5.2 - Trigonometric Functions: Unit Circle Approach
A triangle is a right triangle if one of its angles is a right angle. If q is any acute angle, we
may consider a right triangle having q as one of its angles:
We refer to the sides of the triangle of lengths a , b , and c as the adjacent side, opposite side, and hypotenuse, respectively.
For each q, the six ratios are uniquely determined and hence are functions of q.
They are called the trigonometric functions and are designated as the sine, cosine, tangent, cotangent, secant, and cosecant functions, abbreviated sin, cos, tan, cot, sec, and csc, respectively.
Note:
sin q and csc q are reciprocals of each other, giving us the two identities
Similarly, cos q and sec q are reciprocals of each other, as are tan q and cot q.
Example:
If q is an acute angle and ๐๐๐ q =
! "
, find the values of the trigonometric functions of q.
Example:
Let t be a real number and let ๐ = (โ
$
โ! $
) be the point on the unit circle that
corresponds to t. Find the values of sin t , cos t , tan t , csc t , sec t , and cot t.
Example:
Find the exact values of the six trigonometric functions of ๐ =
& "
Signs of the Trigonometric Functions Let us determine the signs associated with values of the trigonometric functions.
If q is in quadrant II and P ( x , y ) is a point on the terminal side, then x is negative and y is
positive.
Hence, sin q = y / r and csc q = r / y are positive, and the other four trigonometric
functions, which all involve x , are negative. Checking the remaining quadrant in a similar fashion, we obtain the following table: This diagram might be useful for remembering quadrants in which trig functions are positive:
Example: Rewrite the expression in nonradical form โsec!^ ๐ โ 1 if
a) ๐ < ๐ <
!& $ b) & $
Example:
Given that sin ๐ =
! and cos ๐ < 0 , find the exact value of each of the remaining five trigonometric functions. Example: Given that cot ๐ = $ %# and cos ๐ < 0 , find the exact value of each of the remaining five trigonometric functions.
