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7-3 Sine and Cosine (and Tangent) Functions 7-4 Evaluating Sine and Cosine. sin is an abbreviation for sine cos is an abbreviation for cosine tan is an abbreviation for tangent. y. P(x,y ). r. y. 0. x. x. - PowerPoint PPT Presentation
7-3 Sine and Cosine (and Tangent) Functions7-4 Evaluating Sine and Cosine
sin is an abbreviation for sine
cos is an abbreviation for cosine
tan is an abbreviation for tangent
222 ryx 22 yxr
x0
P(x,y)
r
y
x
y
Which of the following represents r in the figure below? (Click on the blue.)
2
xyr
Close. The Pythagorean Theorem would be a good beginning but you will still need to “get r alone.”
You’re kidding right? (xy)/2 represents the area of the triangle!
CORRECT!
r
xsin 0 x,
x
ysin
x0
P(x,y)
r
y
x
y
Which of the following represents sin in the figure below? (Click on the blue.)
r
ysin
Sorry. Does SohCahToa ring a bell? x/r represents cos.
CORRECT! Well done.
Sorry. Does Some Old Hippy Caught Another Hippy Tripping on Acid sound familiar? y/x represents tan.
r
xcos
r
ycos
x0
P(x,y)
r
y
x
y
Which of the following represents cos in the figure below? (Click on the blue.)
0 x,x
ycos
CORRECT! Yeah! Sorry. Wrong ratio.Oops! Try something else.
0 x,x
ytan
r
ytan
x0
P(x,y)
r
y
x
y
Which of the following represents tan in the figure below? (Click on the blue.)
r
xtan
CORRECT! Yeah! Try again.Try again.
x0
P(x,y)
r
y
x
y
In your notes, please copy this figure and the following three ratios:
r
ysin
r
xcos
0 x,x
ytan
222 ryx 22 yxr
0
P(x,y)
r
x
y
r
ysin
r
xcos
0 x,x
ytan
A few key points to write in your notebook:
• P(x,y) can lie in any quadrant.
• Since the hypotenuse r, represents distance, the value of r is always positive.
• The equation x2 + y2 = r2 represents the equation of a circle with its center at the origin and a radius of length r.
• The trigonometric ratios still apply but you will need to pay attention to the +/– sign of each.
Example: If the terminal ray of an angle in standard position passes through (–3, 2), find sin and cos .
13
132
13
13
13
2
13
2
r
ysin
13
133
13
13
13
3
13
3
r
xcos
1323
23
22
r)(r
yx
You try this one in your notebook: If the terminal ray of an angle in standard position passes through (–3, –4), find sin and cos .
(–3,2)
r –3
2
5
35
4
cos
sinCheck Answer
12
144
144
16925
135
2
2
222
x
x
x
x
)(x
13
5
13
5
r
y
r
ysin
13
12
r
xcos
Example: If is a fourth-quadrant angle and sin = –5/13, find cos .
13–5
x
Since is in quadrant IV, the coordinate signs will be (+x, –y), therefore x = +12.
Example: If is a second quadrant angle and cos = –7/25, find sin .
25
24sinCheck Answer
x0
P(–x,y)
r
y
0
P(–x, –y)
rx
y
P(x,y)
0
r
x
y
0
P(x, –y)r
x
y
Determine the signs of sin , cos , and tan according to quadrant. Quadrant II is completed for you. Repeat the process for quadrants I, III, and IV. Hint: r is always positive; look at the red P coordinate to determine the sign of x and y.
(neg)
(neg)
(pos)
II Quadrant
x
ytan
r
xcos
r
ysin
x
ytan
r
xcos
r
ysin
III Quadrant
x
ytan
r
xcos
r
ysin
I Quadrant
x
ytan
r
xcos
r
ysin
IV Quadrant
y
x
AllSine
Tangent Cosine
Check your answers according to the chart below:
•All are positive in I.
•Only sine is positive in II.
•Only tangent is positive in III.
•Only cosine is positive in IV.
y
x
AllStudents
Take Calculus
A handy pneumonic to help you remember! Write it in your notes!
x0
P(–x,y)
r
y
0
P(–x, –y)
r
x
y
P(x,y)
0
r
x
y
0
P(x, –y)r
x
y
Let be an angle in standard position. The reference angle associated with is the acute angle formed by the terminal side of and the x-axis.
1. Find the reference angle.
2. Determine the sign by noting the quadrant.
3. Evaluate and apply the sign.
180
180
2360
Example: Find the reference angle for = 135.
You try it: Find the reference angle for = 5/3.
You try it: Find the reference angle for = 870.
4535180
180
:II quadrant in is 135 Since
3
Check Answer
Check Answer
30
Give each of the following in terms of the cosine of a reference angle:
Example: cos 160The angle =160 is in Quadrant II; cosine is negative in Quadrant II (refer back to All Students Take Calculus pneumonic). The reference angle in Quadrant II is as follows: =180 – or =180 – 160 = 20. Therefore: cos 160 = –cos 20
You try some:
•cos 182
•cos (–100)
•cos 365
2cosCheck Answer
80cosCheck Answer
5cosCheck Answer
Try some sine problems now: Give each of the following in terms of the sine of a reference angle:
•sin 170
•sin 330
•sin (–15)
•sin 400
10sinCheck Answer
30sinCheck Answer
15sinCheck Answer
40sinCheck Answer
(degrees) (radians) sin cos
0 0 0 1
30 6
2
1
2
3
45
60
90 2
1 0
Can you complete this chart?
45
45
2
1
1
60
30
3
1
260
30
21
3
2
330
2
130
213
r
xcos
r
ysin
r,y,x
Check your work!!!!!! Write this table in your notes!
(degrees) (radians) sin cos
0 0 0 1
30 6
2
1
2
3
45 4
2
2
2
2
60 3
2
3
2
1
90 2
1 0
Give the exact value in simplest radical form.
Example: sin 225
Determine the sign: This angle is in Quadrant III where sine isnegative. Find the reference angle for an angle in Quadrant III: = – 180 or = 225 – 180 = 45. Therefore:
(degrees) (radians) sin cos
0 0 0 1
30 6
2
1
2
3
45 4
2
2
2
2
60 3
2
3
2
1
90 2
1 0
2
245225 sinsin
You try some: Give the exact value in simplest radical form:
•sin 45
•sin 135
•sin 225
•cos (–30)
•cos 330
•sin 7/6
•cos /4
2
2Check Answer
2
2Check Answer
2
2Check Answer
2
3Check Answer
2
3Check Answer
2
1Check Answer
2
2Check Answer
Homework: Page 279-280, #1, 3, 11, 13, 15, 17