Trigonometry Basic Flashcards
1
Q
A
2
Q
Define sine in a right triangle.
A
sin(θ) = opposite / hypotenuse.
3
Q
Define cosine in a right triangle.
A
cos(θ) = adjacent / hypotenuse.
4
Q
Define tangent in a right triangle.
A
tan(θ) = opposite / adjacent.
5
Q
What are the three primary trig ratios?
A
Sine, Cosine, Tangent.
6
Q
In a right triangle, if sin(θ)=1, what is θ?
A
θ = 90° (or π/2 radians).
7
Q
Evaluate sin(0°).
A
0.
8
Q
Evaluate cos(0°).
A
1.
9
Q
Evaluate tan(0°).
A
0.
10
Q
Which trig function is undefined at 90°?
A
tan(90°).
11
Q
List the sides for a 45°-45°-90° right triangle.
A
Sides: x, x, x√2 (x is each leg, x√2 is hypotenuse).
12
Q
List the sides for a 30°-60°-90° triangle.
A
Sides: x (opposite 30°), x√3 (opposite 60°), 2x (hypotenuse).
13
Q
Evaluate sin(30°).
A
1/2.
14
Q
Evaluate cos(30°).
A
√3/2.
15
Q
Evaluate tan(30°).
A
1/√3 or √3/3.
16
Q
Evaluate sin(45°).
A
√2/2.
17
Q
Evaluate cos(45°).
A
√2/2.
18
Q
Evaluate tan(45°).
A
1.
19
Q
Evaluate sin(60°).
A
√3/2.
20
Q
Evaluate cos(60°).
A
1/2.
21
Q
Evaluate tan(60°).
A
√3.
22
Q
Evaluate sin(90°).
A
1.
23
Q
Evaluate cos(90°).
A
0.
24
Q
What is the Pythagorean identity involving sin and cos?
A
sin²θ + cos²θ = 1.
25
Rewrite tanθ in terms of sin and cos.
tanθ = sinθ / cosθ.
26
What is 1 + tan²θ equal to?
sec²θ.
27
What is 1 + cot²θ equal to?
csc²θ.
28
Define secθ in a right triangle.
secθ = hypotenuse / adjacent.
29
Define cscθ in a right triangle.
cscθ = hypotenuse / opposite.
30
What is the reciprocal of sinθ?
cscθ.
31
What is the reciprocal of cosθ?
secθ.
32
What is the reciprocal of tanθ?
cotθ.
33
If sinθ=opposite/hypotenuse, then cosθ=?
adjacent/hypotenuse.
34
If tanθ=opposite/adjacent, then cotθ=?
adjacent/opposite.
35
Solve for θ if sinθ=1/2 and θ in [0°, 180°].
θ=30° or 150°.
36
Solve for θ if cosθ=0 in [0°,360°].
θ=90° or 270°.
37
In degrees, for which θ does tanθ=1 if θ in [0°,180°)?
θ=45° or 135°.
38
Convert 90° to radians.
π/2.
39
Convert 180° to radians.
π.
40
Convert 360° to radians.
2π.
41
Convert π/6 radians to degrees.
30°.
42
Convert π/4 radians to degrees.
45°.
43
Convert π/3 radians to degrees.
60°.
44
What is the formula for the sine of a sum: sin(α+β)?
sinα cosβ + cosα sinβ.
45
What is the formula for cos(α+β)?
cosα cosβ − sinα sinβ.
46
What is the formula for tan(α+β)?
(tanα + tanβ)/(1 − tanα tanβ).
47
What is sin(α−β)?
sinα cosβ − cosα sinβ.
48
What is cos(α−β)?
cosα cosβ + sinα sinβ.
49
What is tan(α−β)?
(tanα − tanβ)/(1 + tanα tanβ).
50
What is double-angle formula for sin(2θ)?
2 sinθ cosθ.
51
Double-angle formula for cos(2θ) in one version?
cos²θ − sin²θ.
52
Another form of cos(2θ) using sin²θ?
1 − 2 sin²θ.
53
Another form of cos(2θ) using cos²θ?
2 cos²θ − 1.
54
Double-angle formula for tan(2θ)?
(2 tanθ)/(1 − tan²θ).
55
Half-angle formula for sin(θ/2)?
±√[(1 − cosθ)/2].
56
Half-angle formula for cos(θ/2)?
±√[(1 + cosθ)/2].
57
Identity for sin²θ in terms of cos(2θ)?
sin²θ = (1 − cos(2θ))/2.
58
Identity for cos²θ in terms of cos(2θ)?
cos²θ = (1 + cos(2θ))/2.
59
What is the area formula for a triangle using trigonometry: (½)ab sinC?
Area = (1/2)(side a)(side b) sin(included angle).
60
Law of Sines formula?
a/sinA = b/sinB = c/sinC.
61
Law of Cosines formula for c²?
c² = a² + b² − 2ab cosC.
62
Simplify sin²θ + cos²θ.
1.
63
Simplify (tanθ)(cotθ).
1.
64
For angle θ in quadrant I, sinθ>0 or <0?
sinθ>0.
65
For angle θ in quadrant II, cosθ>0 or <0?
cosθ<0.
66
In quadrant III, are sinθ and cosθ both negative?
Yes, both negative.
67
In quadrant IV, sinθ>0 or sinθ<0?
sinθ<0.
68
If cosθ=0.6, sin²θ=?
sin²θ=1−(0.6)²=1−0.36=0.64 so sinθ=±0.8 (depends quadrant).
69
Which quadrant is angle 150° in?
Quadrant II.
70
Which quadrant is angle 270° in?
Quadrant III–IV boundary (actually exactly on negative y-axis).
71
What is sin(π/2) in radians?
1.
72
What is tan(π/4) in radians?
1.
73
What is cos(π/3) in radians?
1/2.
74
Convert 120° to radians.
2π/3.
75
Convert 270° to radians.
3π/2.
76
Compute sin(π) in exact form.
0.
77
Compute cos(π) in exact form.
−1.
78
Compute tan(π/6) exactly.
1/√3 or √3/3.
79
Compute sec(π/3) exactly.
sec(π/3)=1/cos(π/3)=1/(1/2)=2.
80
Compute csc(π/2) exactly.
1/sin(π/2)=1/1=1.
81
Compute cot(π/4) exactly.
1/tan(π/4)=1/1=1.
82
If sinθ=3/5 and θ in quadrant I, find cosθ.
4/5 (from 3-4-5 triangle).
83
If sinθ=4/5 in quadrant II, what is cosθ?
−3/5 (since cos is negative in quadrant II).
84
If tanθ=2, find θ in degrees for the principal value (0°<θ<90°).
θ≈63.435° (approx).
85
Solve sinθ=0.5 in degrees with 0°≤θ<360°.
θ=30° or 150°.
86
Solve cosθ=−√2/2 in [0°,360°).
θ=135° or 225°.
87
Solve tanθ=√3 in degrees [0°,360°).
θ=60°, 240°.
88
What angle has sinθ=√2/2 in [0°,360°)?
θ=45° or 135°.
89
What angle has tanθ=−1 in [0°,360°)?
θ=135°, 315°.
90
Simplify sin(π−θ).
sin(π−θ)=sinθ.
91
Simplify cos(π−θ).
cos(π−θ)=−cosθ.
92
Express sin(−θ) in terms of sinθ.
sin(−θ)=−sinθ (odd function).
93
Express cos(−θ) in terms of cosθ.
cos(−θ)=cosθ (even function).
94
Express tan(−θ) in terms of tanθ.
tan(−θ)=−tanθ (odd function).
95
If angle A=π/2−B, rewrite sinA in terms of B.
sinA=cosB.
96
Likewise, cos(π/2−B)=?
sinB.
97
For a right triangle with sides a=6, c=10, find b if c is hypotenuse.
b=√(c²−a²)=√(100−36)=√64=8.
98
In a circle, define arc length formula for central angle θ in radians.
Arc length = rθ.
99
Name the ratio that remains constant in all similar right triangles for a given angle.
sinθ, cosθ, or tanθ (the trigonometric ratio).
100
Is sinθ>cosθ always for 0<θ<90°?
Not always. It's true if θ>45°. For θ<45°, cosθ>sinθ.
101
If tanθ=opposite/adjacent, then name the related Pythagorean identity for tanθ, secθ.
1 + tan²θ = sec²θ.
102
Definition of sinθ in a right triangle
sinθ = opposite / hypotenuse
103
Definition of cosθ in a right triangle
cosθ = adjacent / hypotenuse
104
Definition of tanθ in a right triangle
tanθ = opposite / adjacent
105
Definition of cscθ
cscθ = 1 / sinθ
106
Definition of secθ
secθ = 1 / cosθ
107
Definition of cotθ
cotθ = 1 / tanθ
108
Pythagorean identity #1
sin²θ + cos²θ = 1
109
Pythagorean identity #2
1 + tan²θ = sec²θ
110
Pythagorean identity #3
1 + cot²θ = csc²θ
111
Quotient identity for tanθ
tanθ = sinθ / cosθ
112
Quotient identity for cotθ
cotθ = cosθ / sinθ
113
Sum identity for sin(α + β)
sin(α + β) = sinα cosβ + cosα sinβ
114
Sum identity for cos(α + β)
cos(α + β) = cosα cosβ − sinα sinβ
115
Sum identity for tan(α + β)
tan(α + β) = [tanα + tanβ] / [1 − tanα tanβ]
116
Difference identity for sin(α − β)
sin(α − β) = sinα cosβ − cosα sinβ
117
Difference identity for cos(α − β)
cos(α − β) = cosα cosβ + sinα sinβ
118
Difference identity for tan(α − β)
tan(α − β) = [tanα − tanβ] / [1 + tanα tanβ]
119
Double-angle formula for sin(2θ)
2 sinθ cosθ
120
Double-angle formula for cos(2θ), version 1
cos(2θ) = cos²θ − sin²θ
121
Double-angle formula for cos(2θ), version 2
cos(2θ) = 1 − 2 sin²θ
122
Double-angle formula for cos(2θ), version 3
cos(2θ) = 2 cos²θ − 1
123
Double-angle formula for tan(2θ)
tan(2θ) = (2 tanθ) / (1 − tan²θ)
124
Half-angle formula for sin(θ/2)
±√[(1 − cosθ)/2]
125
Half-angle formula for cos(θ/2)
±√[(1 + cosθ)/2]
126
Half-angle formula for tan(θ/2)
tan(θ/2) = sinθ / (1 + cosθ), or (1−cosθ) / sinθ
127
Identity: sin²θ in terms of cos(2θ)
sin²θ = [1 − cos(2θ)] / 2
128
Identity: cos²θ in terms of cos(2θ)
cos²θ = [1 + cos(2θ)] / 2
129
Cofunction identity: sin(π/2 − x)
cos x
130
Cofunction identity: cos(π/2 − x)
sin x
131
Cofunction identity: tan(π/2 − x)
cot x
132
Cofunction identity: csc(π/2 − x)
sec x
133
Cofunction identity: sec(π/2 − x)
csc x
134
Cofunction identity: cot(π/2 − x)
tan x
135
Angle sum identity for sin(A + B)
sinA cosB + cosA sinB (restate, easy recall)
136
Angle sum identity for cos(A + B)
cosA cosB − sinA sinB
137
Express cscθ using a right triangle perspective
cscθ = hypotenuse / opposite
138
Express secθ using a right triangle perspective
secθ = hypotenuse / adjacent
139
Express cotθ using a right triangle perspective
cotθ = adjacent / opposite
140
Identity: tan(θ) × cot(θ)
1
141
Identity: sin(−θ)
−sinθ (odd function)
142
Identity: cos(−θ)
cosθ (even function)
143
Identity: tan(−θ)
−tanθ (odd function)
144
Sum-to-product: sinA + sinB
2 sin[(A+B)/2] cos[(A−B)/2]
145
Sum-to-product: sinA − sinB
2 cos[(A+B)/2] sin[(A−B)/2]
146
Sum-to-product: cosA + cosB
2 cos[(A+B)/2] cos[(A−B)/2]
147
Sum-to-product: cosA − cosB
−2 sin[(A+B)/2] sin[(A−B)/2]
148
Product-to-sum: sinA sinB
½[cos(A−B) − cos(A+B)]
149
Product-to-sum: cosA cosB
½[cos(A−B) + cos(A+B)]
150
Product-to-sum: sinA cosB
½[sin(A+B) + sin(A−B)]
151
Product-to-sum: cosA sinB
½[sin(A+B) − sin(A−B)]
152
On the unit circle, what does an angle θ correspond to in coordinates?
Point (cosθ, sinθ).
153
How long is the radius in the unit circle?
1 (by definition).
154
For an angle θ in standard position, where does it start and how does it rotate?
Starts along positive x-axis (0° or 0 radians) and rotates counterclockwise.
155
What is the range of θ to complete one full revolution?
0° ≤ θ < 360°, or 0 ≤ θ < 2π (radians).
156
If a point on the unit circle has coordinates (x, y), how do we interpret x and y?
x = cosθ, y = sinθ.
157
On the unit circle, what's cos(0°) and sin(0°)?
(1, 0).
158
Coordinates for θ=30°?
(√3/2, 1/2).
159
Coordinates for θ=45°?
(√2/2, √2/2).
160
Coordinates for θ=60°?
(1/2, √3/2).
161
Coordinates for θ=90°?
(0, 1).
162
Coordinates for θ=120°?
(−1/2, √3/2).
163
Coordinates for θ=135°?
(−√2/2, √2/2).
164
Coordinates for θ=150°?
(−√3/2, 1/2).
165
Coordinates for θ=180°?
(−1, 0).
166
At θ=180°, what is sin(180°) and cos(180°)?
sin=0, cos=−1.
167
Coordinates for θ=210°?
(−√3/2, −1/2).
168
Coordinates for θ=225°?
(−√2/2, −√2/2).
169
Coordinates for θ=240°?
(−1/2, −√3/2).
170
Coordinates for θ=270°?
(0, −1).
171
At θ=270°, what is sin(270°) and cos(270°)?
sin=−1, cos=0.
172
Coordinates for θ=300°?
(1/2, −√3/2).
173
Coordinates for θ=315°?
(√2/2, −√2/2).
174
Coordinates for θ=330°?
(√3/2, −1/2).
175
Coordinates for θ=360°?
(1, 0).
176
At θ=360°, what are sin(360°) and cos(360°)?
sin=0, cos=1.
177
Coordinates for θ=0 radians?
(1, 0).
178
Coordinates for θ=π/6 (~30°)?
(√3/2, 1/2).
179
Coordinates for θ=π/4 (~45°)?
(√2/2, √2/2).
180
Coordinates for θ=π/3 (~60°)?
(1/2, √3/2).
181
Coordinates for θ=π/2 (~90°)?
(0, 1).
182
Coordinates for θ=2π/3 (~120°)?
(−1/2, √3/2).
183
Coordinates for θ=3π/4 (~135°)?
(−√2/2, √2/2).
184
Coordinates for θ=5π/6 (~150°)?
(−√3/2, 1/2).
185
Coordinates for θ=π (~180°)?
(−1, 0).
186
At θ=π, sin(π), cos(π)?
sin=0, cos=−1.
187
In Quadrant I, sign of sin and cos?
Both positive (sin>0, cos>0).
188
In Quadrant II, sign of sin and cos?
sin>0, cos<0.
189
In Quadrant III, sign of sin and cos?
Both negative (sin<0, cos<0).
190
In Quadrant IV, sign of sin and cos?
sin<0, cos>0.
191
What is a reference angle in the unit circle?
Acute angle formed with the x-axis, used to find trig values’ magnitude.
192
Reference angle for 150°?
180° − 150°=30°.
193
Reference angle for 210°?
210° − 180°=30°.
194
Reference angle for 315°?
360° − 315°=45°.
195
Reference angle for 5π/6?
π − 5π/6 = π/6.
196
Reference angle for 7π/4?
2π − 7π/4 = π/4.
197
Convert 90° to radians.
π/2.
198
Convert 180° to radians.
π.
199
Convert 270° to radians.
3π/2.
200
Convert 360° to radians.
2π.
201
Convert π/6 radians to degrees.
30°.
202
sin(45°) or sin(π/4)?
√2/2.
203
cos(60°) or cos(π/3)?
1/2.
204
sin(180°) or sin(π)?
0.
205
cos(270°) or cos(3π/2)?
0.
206
sin(360°) or sin(2π)?
0.
207
tan(45°) or tan(π/4)?
1.
208
tan(90°) or tan(π/2) – is it defined?
No, it's undefined (vertical asymptote).
209
tan(180°) or tan(π)?
0.
210
sec(0°) or sec(0)?
1/cos(0)=1/1=1.
211
csc(90°) or csc(π/2)?
1/sin(π/2)=1/1=1.
212
Coordinates for θ=−30°?
(√3/2, −1/2).
213
Coordinates for θ=−45°?
(√2/2, −√2/2).
214
Coordinates for θ=−90°?
(0, −1).
215
Coordinates for θ=−π/6?
(√3/2, −1/2).
216
Coordinates for θ=−π/2?
(0, −1).
217
Arc length formula for angle θ (in radians) on a unit circle?
Arc length = θ × radius = θ.
218
If radius=1, how many radians is one full revolution?
2π.
219
What is the measure in degrees for 1 radian, approximately?
≈57.2958°.
220
Convert 3π/2 radians to degrees.
270°.
221
If an object moves at 2π radians per second on a unit circle, how many rotations per second?
1 rotation per second.
222
How does the sine function relate to the y-coordinate on the unit circle?
sinθ = y.
223
How does the cosine function relate to the x-coordinate?
cosθ = x.
224
When x=0 on the unit circle, what angles are possible in [0, 2π)?
θ=π/2 or θ=3π/2.
225
When y=0, what angles in [0, 2π)?
θ=0, π, 2π.
226
The function y=cosθ starts at cos(0)=1 on the unit circle. T/F?
True.
227
In quadrant II, what is the sign of sinθ?
Positive (y>0).
228
In quadrant II, sign of cosθ?
Negative (x<0).
229
In quadrant IV, sign of sinθ?
Negative.
230
In quadrant IV, sign of cosθ?
Positive.
231
At 225° (or 5π/4), is sinθ positive or negative?
Negative, quadrant III.
232
What is sin(210°) or sin(7π/6) on the unit circle?
−1/2.
233
What is cos(300°) or cos(5π/3) on the unit circle?
1/2.
234
If cosθ=√3/2 in quadrant I, which angle?
θ=30° or π/6.
235
If sinθ=−√2/2 in quadrant IV, which angle?
θ=315° or 7π/4.
236
If tanθ is positive, which quadrants does θ land in?
Quadrant I or III.
237
If θ=π/2, how many degrees is that?
90°.
238
If θ=2π/3, how many degrees is that?
120°.
239
If θ=4π/3, how many degrees is that?
240°.
240
If θ=5π/4, how many degrees is that?
225°.
241
θ=11π/6 is how many degrees?
330°.
