Download STUDY GUIDE. SECTION 0.5

STUDY GUIDE. SECTION 0.5
How to use this section
• Review this material before you study Sections 3.5 and 3.6.
• Review how the radian measure of angles is defined and how it relates to degrees.
• Be sure you know the geometric definitions of the sine, cosine and tangent functions and how
these determine the shapes of the curves y = sin x, y = cos x, and y = tan x.
• Be sure you know the definitions of the secant cosecant, and cotangent functions in terms of
sines, cosines, and tangents, and how the shapes of the curves y = x, y = csc x, and y = cot x
are determined by the graphs of sine, cosine, and tangent.
• Be sure you understand the definitions of
(a) y = sin–1 x as the inverse of y = sin x with domain restricted to [ 21 π, 21 π]
(b) y = cos–1 x as the inverse of y = cos x with domain restricted to [0, π]
(c) y = tan–1 x as the inverse of y = tan x with domain restricted to ( 21 π, 12 π).
• Know how the graphs of y = sin–1 x, y = cos–1 x and y = tan–1 x are obtained from the graphs
of y = sin x, y = cos x, and y = tan x.
Techniques
• To recall the exact value of sin 14 π or cos 14 π , draw an isosceles-right triangle with legs of
√
√
length 1. Its acute angles are 14 π and the length of its hypotenuse is 12 + 12 = 2. The sine
and cosine of 41 π equal the length of a leg divided by the length of the hypotenuse, which is
√
1
√ = 21 2.
2
• To recall the exact value of sin x or cos x, with x = 61 π or x = 31 π, draw a right triangle with
one leg of length 1 and hypotenuse of length 2. Because it is half of an isosceles triangle with
sides of length 2, all of whose angles are 13 π, the smallest acute angle in this triangle is 16 π and
√
√
its largest acute angle is 13 π. The length of its other leg is 22 − 12 = 3. Express the sine
of either acute angle in the triangle as the length of the side opposite the angle divided by the
length of the hypotenuse or the cosine as the length of the side adjacent to the angle divided by
the length of the hypotenuse.
• To find the exact value of sin x or cos x, where x is an integer multiple of 14 π, 16 π or 13 π, draw
the angle with a unit circle in a uv-plane. Use the drawing to determine whether the sine or
cosine is positive, negative, or zero. If the sine or cosine is not 0 or ±1, use a right triangle with
angles 14 π or 16 π and 13 π to find the absolute value of the sine or cosine.
• To find the exact value of sin x or cos x with x that is not an integer multiple of 14 π, 61 π or 13 π,
draw the angle with a unit circle in a uv-plane. Use the drawing to determine whether the sine
or cosine is positive or negative. Then use an inverse sine, inverse cosine, or inverse tangent with
a right triangle to give its absolute value.
• If a right triangle has an acute angle x and hypotenuse of length c, you can find the lengths of
its legs with the formulas a = c cos x and b = c sin x.
• If a right triangle has an acute angle x and a adjacent leg of length a, then the length of its
other leg is b = a tan x.
• If you are given the coordinates of a point P , its distance to another point Q, and the angle of
inclination of the line segment P Q, you can find the coordinates of Q by using a sine and cosine,
as in Example 2.
1
p. 2
Section 0.5
• To solve the equation sin x = k for x, draw the horizontal line v = k in an uv-plane and find the
angles whose terminal sides intersect the unit circle on this line.
• To solve the equation cos x = k for x, draw the vertical line u = k in an uv-plane and find the
angles whose terminal sides intersect the unit circle on this line.
• The Law of Cosines and Law of Sines are used with triangles that are not right triangles.
The Law of Cosines relates the lengths of two sides of a triangle and the enclosed angle to
the length of the opposite side.
The Law of Sines relates two angles in a triangle and the length of the side between them to
the length of the opposite side.
Other trigonometric identities
• The definitions using a unit circle show that y = cos x and y = − sin x are periodic of period 2π.
As a consequence, y = sec x = 1/(cos x), and y = csc x = 1/(sin x) are also periodic of period
2π:
cos(x + 2π) = cos x
sin(x + 2π) = sin x
sec(x + 2π) = sec x
csc(x + 2π) = csc x
• The Pythagorean identity
cos2 x + sinx = 1
holds because (cos x, sin x) is a point on the unit circle. The other Pythagorean identities,
1 + tan2 x = sec2 x
cot2 x + 1 = csc2 x
follow by dividing by cos x and sin x, respectively.
• The following identities are also easy to recall from the definitions.
cos(−x) = cos x
sin(−x) = − sin x
cos(x ± π) = − cos x
sin(x ± π) = − sin x
• The last two identities in this list imply that y = tan x and y = cot x are periodic of period π:
tan(x + π) = tan x
cot(x + π) = cot x
• Other trigonometric identities will be discussed as they are needed later in the text.
p. 3 p. 3
Section 0.5
Tips
• An angle of one radian at the center of a circle of radius r subtends an arc of length r on the
circle. Consequently, one radian is slightly less than 60◦ .
• Recall that powers of trigonometric functions are written sinn x, cosn x, etc., instead of
(sin x)n , (cos x)n , etc.
• If you use a calculator or computer with trigonometric functions be sure that the instrument is
in radian, rather than degree, mode.