Example 1
Compute the sine and cosine of π/3.
Solution 1
We sketch the terminal radius and associated triangle (see Fig. 1.33 ). This is a 30–60–90 triangle whose sides have ratios . Thus
Likewise,
It follows that
and
You Try It: The cosine of a certain angle is 2/3. The angle lies in the fourth quadrant. What is the sine of the angle?
Math Note: Notice that if θ is an angle then θ and θ + 2π have the same terminal radius and the same terminal point (for adding 2 just adds one more trip around the circle—look at Fig. 1.34).
As a result,
sin θ = x = sin( θ + 2π)
and
cos θ = y = cos( θ + 2π).
Associated Principle Angle
We say that the sine and cosine functions have period 2π: the functions repeat themselves every 2π units.
In practice, when we calculate the trigonometric functions of an angle θ , we reduce it by multiples of 2π so that we can consider an equivalent angle θ′ , called the associated principal angle, satisfying 0 ≤ θ′ < 2π. For instance,
15π/2 has associated principal angle
3π/2 (since 15π/2 − 3π/2 = 3·2π )
and
− 10π/3 has associated principal angle
2π/3 (since − 10π/3 − 2π/3 = − 12π/3 = −2 ·2π ).
You Try It: What are the principal angles associated with 7π, 11π/2, 8π/3, −14π/5, −16π/7?
Sine, Cosine, and Right Triangle Trigonometry
What does the concept of angle and sine and cosine that we have presented here have to do with the classical notion using triangles? Notice that any angle θ such that 0 ≤ θ < π/2 has associated to it a right triangle in the first quadrant, with vertex on the unit circle, such that the base is the segment connecting (0, 0) to ( x , 0) and the height is the segment connecting ( x , 0) to ( x , y ). See Fig. 1.35.
Then
and
Thus, for angles θ between 0 and π/2, the new definition of sine and cosine using the unit circle is clearly equivalent to the classical definition using adjacent and opposite sides and the hypotenuse. For other angles θ , the classical approach is to reduce to this special case by subtracting multiples of π/2. Our approach using the unit circle is considerably clearer because it makes the signatures of sine and cosine obvious.
Tangent, Cotangent, Secant, and Cosecant
Besides sine and cosine, there are four other trigonometric functions:
Whereas sine and cosine have domain the entire real line, we notice that tan θ and sec θ are undefined at odd multiples of π/2 (because cosine will vanish there) and cot θ and csc θ are undefined at even multiples of π/2 (because sine will vanish there). The graphs of the six trigonometric functions are shown in Fig. 1.36.
Example 1
Compute all the trigonometric functions for the angle θ = 11π/4.
Solution 1
We first notice that the principal associated angle is 3π/4, so we deal with that angle. Figure 1.37 shows that the triangle associated to this angle is an isosceles right triangle with hypotenuse 1.
Therefore and . It follows that
Similar calculations allow us to complete the following table for the values of the trigonometric functions at the principal angles which are multiples of π/6 or π/4.
Fundamental Properties for Trigonometry Functions
(1) For any number θ ,
(sin θ ) ^{2} + (cos θ ) ^{2} = 1.
(2) For any number θ ,
−1 ≤ cos θ ≤ 1 and −1 ≤ sin θ ≤ 1.
Besides properties (1) and (2) above, there are certain identities which are fundamental to our study of the trigonometric functions. Here are the principal ones:
Example 1
Prove identity number (3).
Solution 1
We have
You Try It: Use identities (11) and (12) to calculate cos(π/12) and sin(π/12).
Find practice problems and solutions for these concepts at: Calculus Basics Practice Test.
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