Trigonometry Basic Flashcards
Define sine in a right triangle.
sin(θ) = opposite / hypotenuse.
Define cosine in a right triangle.
cos(θ) = adjacent / hypotenuse.
Define tangent in a right triangle.
tan(θ) = opposite / adjacent.
What are the three primary trig ratios?
Sine, Cosine, Tangent.
In a right triangle, if sin(θ)=1, what is θ?
θ = 90° (or π/2 radians).
Evaluate sin(0°).
0.
Evaluate cos(0°).
1.
Evaluate tan(0°).
0.
Which trig function is undefined at 90°?
tan(90°).
List the sides for a 45°-45°-90° right triangle.
Sides: x, x, x√2 (x is each leg, x√2 is hypotenuse).
List the sides for a 30°-60°-90° triangle.
Sides: x (opposite 30°), x√3 (opposite 60°), 2x (hypotenuse).
Evaluate sin(30°).
1/2.
Evaluate cos(30°).
√3/2.
Evaluate tan(30°).
1/√3 or √3/3.
Evaluate sin(45°).
√2/2.
Evaluate cos(45°).
√2/2.
Evaluate tan(45°).
1.
Evaluate sin(60°).
√3/2.
Evaluate cos(60°).
1/2.
Evaluate tan(60°).
√3.
Evaluate sin(90°).
1.
Evaluate cos(90°).
0.
What is the Pythagorean identity involving sin and cos?
sin²θ + cos²θ = 1.
Rewrite tanθ in terms of sin and cos.
tanθ = sinθ / cosθ.
What is 1 + tan²θ equal to?
sec²θ.
What is 1 + cot²θ equal to?
csc²θ.
Define secθ in a right triangle.
secθ = hypotenuse / adjacent.
Define cscθ in a right triangle.
cscθ = hypotenuse / opposite.
What is the reciprocal of sinθ?
cscθ.
What is the reciprocal of cosθ?
secθ.
What is the reciprocal of tanθ?
cotθ.
If sinθ=opposite/hypotenuse, then cosθ=?
adjacent/hypotenuse.
If tanθ=opposite/adjacent, then cotθ=?
adjacent/opposite.
Solve for θ if sinθ=1/2 and θ in [0°, 180°].
θ=30° or 150°.
Solve for θ if cosθ=0 in [0°,360°].
θ=90° or 270°.
In degrees, for which θ does tanθ=1 if θ in [0°,180°)?
θ=45° or 135°.
Convert 90° to radians.
π/2.
Convert 180° to radians.
π.
Convert 360° to radians.
2π.
Convert π/6 radians to degrees.
30°.
Convert π/4 radians to degrees.
45°.
Convert π/3 radians to degrees.
60°.
What is the formula for the sine of a sum: sin(α+β)?
sinα cosβ + cosα sinβ.
What is the formula for cos(α+β)?
cosα cosβ − sinα sinβ.
What is the formula for tan(α+β)?
(tanα + tanβ)/(1 − tanα tanβ).
What is sin(α−β)?
sinα cosβ − cosα sinβ.
What is cos(α−β)?
cosα cosβ + sinα sinβ.
What is tan(α−β)?
(tanα − tanβ)/(1 + tanα tanβ).
What is double-angle formula for sin(2θ)?
2 sinθ cosθ.
Double-angle formula for cos(2θ) in one version?
cos²θ − sin²θ.
Another form of cos(2θ) using sin²θ?
1 − 2 sin²θ.
Another form of cos(2θ) using cos²θ?
2 cos²θ − 1.
Double-angle formula for tan(2θ)?
(2 tanθ)/(1 − tan²θ).
Half-angle formula for sin(θ/2)?
±√[(1 − cosθ)/2].
Half-angle formula for cos(θ/2)?
±√[(1 + cosθ)/2].
Identity for sin²θ in terms of cos(2θ)?
sin²θ = (1 − cos(2θ))/2.
Identity for cos²θ in terms of cos(2θ)?
cos²θ = (1 + cos(2θ))/2.
What is the area formula for a triangle using trigonometry: (½)ab sinC?
Area = (1/2)(side a)(side b) sin(included angle).
Law of Sines formula?
a/sinA = b/sinB = c/sinC.
Law of Cosines formula for c²?
c² = a² + b² − 2ab cosC.
Simplify sin²θ + cos²θ.
1.
Simplify (tanθ)(cotθ).
1.
For angle θ in quadrant I, sinθ>0 or <0?
sinθ>0.
For angle θ in quadrant II, cosθ>0 or <0?
cosθ<0.
In quadrant III, are sinθ and cosθ both negative?
Yes, both negative.
In quadrant IV, sinθ>0 or sinθ<0?
sinθ<0.
If cosθ=0.6, sin²θ=?
sin²θ=1−(0.6)²=1−0.36=0.64 so sinθ=±0.8 (depends quadrant).
Which quadrant is angle 150° in?
Quadrant II.
Which quadrant is angle 270° in?
Quadrant III–IV boundary (actually exactly on negative y-axis).
What is sin(π/2) in radians?
1.
What is tan(π/4) in radians?
1.
What is cos(π/3) in radians?
1/2.
Convert 120° to radians.
2π/3.
Convert 270° to radians.
3π/2.
Compute sin(π) in exact form.
0.
Compute cos(π) in exact form.
−1.
Compute tan(π/6) exactly.
1/√3 or √3/3.
Compute sec(π/3) exactly.
sec(π/3)=1/cos(π/3)=1/(1/2)=2.
Compute csc(π/2) exactly.
1/sin(π/2)=1/1=1.
Compute cot(π/4) exactly.
1/tan(π/4)=1/1=1.
If sinθ=3/5 and θ in quadrant I, find cosθ.
4/5 (from 3-4-5 triangle).
If sinθ=4/5 in quadrant II, what is cosθ?
−3/5 (since cos is negative in quadrant II).
If tanθ=2, find θ in degrees for the principal value (0°<θ<90°).
θ≈63.435° (approx).
Solve sinθ=0.5 in degrees with 0°≤θ<360°.
θ=30° or 150°.
Solve cosθ=−√2/2 in [0°,360°).
θ=135° or 225°.
Solve tanθ=√3 in degrees [0°,360°).
θ=60°, 240°.
What angle has sinθ=√2/2 in [0°,360°)?
θ=45° or 135°.
What angle has tanθ=−1 in [0°,360°)?
θ=135°, 315°.
Simplify sin(π−θ).
sin(π−θ)=sinθ.
Simplify cos(π−θ).
cos(π−θ)=−cosθ.
Express sin(−θ) in terms of sinθ.
sin(−θ)=−sinθ (odd function).
Express cos(−θ) in terms of cosθ.
cos(−θ)=cosθ (even function).
Express tan(−θ) in terms of tanθ.
tan(−θ)=−tanθ (odd function).
If angle A=π/2−B, rewrite sinA in terms of B.
sinA=cosB.
Likewise, cos(π/2−B)=?
sinB.
For a right triangle with sides a=6, c=10, find b if c is hypotenuse.
b=√(c²−a²)=√(100−36)=√64=8.
In a circle, define arc length formula for central angle θ in radians.
Arc length = rθ.
Name the ratio that remains constant in all similar right triangles for a given angle.
sinθ, cosθ, or tanθ (the trigonometric ratio).
Is sinθ>cosθ always for 0<θ<90°?
Not always. It’s true if θ>45°. For θ<45°, cosθ>sinθ.
If tanθ=opposite/adjacent, then name the related Pythagorean identity for tanθ, secθ.
1 + tan²θ = sec²θ.
Definition of sinθ in a right triangle
sinθ = opposite / hypotenuse
Definition of cosθ in a right triangle
cosθ = adjacent / hypotenuse
Definition of tanθ in a right triangle
tanθ = opposite / adjacent
Definition of cscθ
cscθ = 1 / sinθ
Definition of secθ
secθ = 1 / cosθ
Definition of cotθ
cotθ = 1 / tanθ
Pythagorean identity #1
sin²θ + cos²θ = 1
Pythagorean identity #2
1 + tan²θ = sec²θ
Pythagorean identity #3
1 + cot²θ = csc²θ
Quotient identity for tanθ
tanθ = sinθ / cosθ
Quotient identity for cotθ
cotθ = cosθ / sinθ
Sum identity for sin(α + β)
sin(α + β) = sinα cosβ + cosα sinβ
Sum identity for cos(α + β)
cos(α + β) = cosα cosβ − sinα sinβ
Sum identity for tan(α + β)
tan(α + β) = [tanα + tanβ] / [1 − tanα tanβ]
Difference identity for sin(α − β)
sin(α − β) = sinα cosβ − cosα sinβ
Difference identity for cos(α − β)
cos(α − β) = cosα cosβ + sinα sinβ
Difference identity for tan(α − β)
tan(α − β) = [tanα − tanβ] / [1 + tanα tanβ]
Double-angle formula for sin(2θ)
2 sinθ cosθ
Double-angle formula for cos(2θ), version 1
cos(2θ) = cos²θ − sin²θ
Double-angle formula for cos(2θ), version 2
cos(2θ) = 1 − 2 sin²θ
Double-angle formula for cos(2θ), version 3
cos(2θ) = 2 cos²θ − 1
Double-angle formula for tan(2θ)
tan(2θ) = (2 tanθ) / (1 − tan²θ)
Half-angle formula for sin(θ/2)
±√[(1 − cosθ)/2]
Half-angle formula for cos(θ/2)
±√[(1 + cosθ)/2]
Half-angle formula for tan(θ/2)
tan(θ/2) = sinθ / (1 + cosθ), or (1−cosθ) / sinθ
Identity: sin²θ in terms of cos(2θ)
sin²θ = [1 − cos(2θ)] / 2
Identity: cos²θ in terms of cos(2θ)
cos²θ = [1 + cos(2θ)] / 2
Cofunction identity: sin(π/2 − x)
cos x
Cofunction identity: cos(π/2 − x)
sin x
Cofunction identity: tan(π/2 − x)
cot x
Cofunction identity: csc(π/2 − x)
sec x
Cofunction identity: sec(π/2 − x)
csc x
Cofunction identity: cot(π/2 − x)
tan x
Angle sum identity for sin(A + B)
sinA cosB + cosA sinB (restate, easy recall)
Angle sum identity for cos(A + B)
cosA cosB − sinA sinB
Express cscθ using a right triangle perspective
cscθ = hypotenuse / opposite
Express secθ using a right triangle perspective
secθ = hypotenuse / adjacent
Express cotθ using a right triangle perspective
cotθ = adjacent / opposite
Identity: tan(θ) × cot(θ)
1
Identity: sin(−θ)
−sinθ (odd function)
Identity: cos(−θ)
cosθ (even function)
Identity: tan(−θ)
−tanθ (odd function)
Sum-to-product: sinA + sinB
2 sin[(A+B)/2] cos[(A−B)/2]
Sum-to-product: sinA − sinB
2 cos[(A+B)/2] sin[(A−B)/2]
Sum-to-product: cosA + cosB
2 cos[(A+B)/2] cos[(A−B)/2]
Sum-to-product: cosA − cosB
−2 sin[(A+B)/2] sin[(A−B)/2]
Product-to-sum: sinA sinB
½[cos(A−B) − cos(A+B)]
Product-to-sum: cosA cosB
½[cos(A−B) + cos(A+B)]
Product-to-sum: sinA cosB
½[sin(A+B) + sin(A−B)]
Product-to-sum: cosA sinB
½[sin(A+B) − sin(A−B)]
On the unit circle, what does an angle θ correspond to in coordinates?
Point (cosθ, sinθ).
How long is the radius in the unit circle?
1 (by definition).
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.
What is the range of θ to complete one full revolution?
0° ≤ θ < 360°, or 0 ≤ θ < 2π (radians).
If a point on the unit circle has coordinates (x, y), how do we interpret x and y?
x = cosθ, y = sinθ.
On the unit circle, what’s cos(0°) and sin(0°)?
(1, 0).
Coordinates for θ=30°?
(√3/2, 1/2).
Coordinates for θ=45°?
(√2/2, √2/2).
Coordinates for θ=60°?
(1/2, √3/2).
Coordinates for θ=90°?
(0, 1).
Coordinates for θ=120°?
(−1/2, √3/2).
Coordinates for θ=135°?
(−√2/2, √2/2).
Coordinates for θ=150°?
(−√3/2, 1/2).
Coordinates for θ=180°?
(−1, 0).
At θ=180°, what is sin(180°) and cos(180°)?
sin=0, cos=−1.
Coordinates for θ=210°?
(−√3/2, −1/2).
Coordinates for θ=225°?
(−√2/2, −√2/2).
Coordinates for θ=240°?
(−1/2, −√3/2).
Coordinates for θ=270°?
(0, −1).
At θ=270°, what is sin(270°) and cos(270°)?
sin=−1, cos=0.
Coordinates for θ=300°?
(1/2, −√3/2).
Coordinates for θ=315°?
(√2/2, −√2/2).
Coordinates for θ=330°?
(√3/2, −1/2).
Coordinates for θ=360°?
(1, 0).
At θ=360°, what are sin(360°) and cos(360°)?
sin=0, cos=1.
Coordinates for θ=0 radians?
(1, 0).
Coordinates for θ=π/6 (~30°)?
(√3/2, 1/2).
Coordinates for θ=π/4 (~45°)?
(√2/2, √2/2).
Coordinates for θ=π/3 (~60°)?
(1/2, √3/2).
Coordinates for θ=π/2 (~90°)?
(0, 1).
Coordinates for θ=2π/3 (~120°)?
(−1/2, √3/2).
Coordinates for θ=3π/4 (~135°)?
(−√2/2, √2/2).
Coordinates for θ=5π/6 (~150°)?
(−√3/2, 1/2).
Coordinates for θ=π (~180°)?
(−1, 0).
At θ=π, sin(π), cos(π)?
sin=0, cos=−1.
In Quadrant I, sign of sin and cos?
Both positive (sin>0, cos>0).
In Quadrant II, sign of sin and cos?
sin>0, cos<0.
In Quadrant III, sign of sin and cos?
Both negative (sin<0, cos<0).
In Quadrant IV, sign of sin and cos?
sin<0, cos>0.
What is a reference angle in the unit circle?
Acute angle formed with the x-axis, used to find trig values’ magnitude.
Reference angle for 150°?
180° − 150°=30°.
Reference angle for 210°?
210° − 180°=30°.
Reference angle for 315°?
360° − 315°=45°.
Reference angle for 5π/6?
π − 5π/6 = π/6.
Reference angle for 7π/4?
2π − 7π/4 = π/4.
Convert 90° to radians.
π/2.
Convert 180° to radians.
π.
Convert 270° to radians.
3π/2.
Convert 360° to radians.
2π.
Convert π/6 radians to degrees.
30°.
sin(45°) or sin(π/4)?
√2/2.
cos(60°) or cos(π/3)?
1/2.
sin(180°) or sin(π)?
0.
cos(270°) or cos(3π/2)?
0.
sin(360°) or sin(2π)?
0.
tan(45°) or tan(π/4)?
1.
tan(90°) or tan(π/2) – is it defined?
No, it’s undefined (vertical asymptote).
tan(180°) or tan(π)?
0.
sec(0°) or sec(0)?
1/cos(0)=1/1=1.
csc(90°) or csc(π/2)?
1/sin(π/2)=1/1=1.
Coordinates for θ=−30°?
(√3/2, −1/2).
Coordinates for θ=−45°?
(√2/2, −√2/2).
Coordinates for θ=−90°?
(0, −1).
Coordinates for θ=−π/6?
(√3/2, −1/2).
Coordinates for θ=−π/2?
(0, −1).
Arc length formula for angle θ (in radians) on a unit circle?
Arc length = θ × radius = θ.
If radius=1, how many radians is one full revolution?
2π.
What is the measure in degrees for 1 radian, approximately?
≈57.2958°.
Convert 3π/2 radians to degrees.
270°.
If an object moves at 2π radians per second on a unit circle, how many rotations per second?
1 rotation per second.
How does the sine function relate to the y-coordinate on the unit circle?
sinθ = y.
How does the cosine function relate to the x-coordinate?
cosθ = x.
When x=0 on the unit circle, what angles are possible in [0, 2π)?
θ=π/2 or θ=3π/2.
When y=0, what angles in [0, 2π)?
θ=0, π, 2π.
The function y=cosθ starts at cos(0)=1 on the unit circle. T/F?
True.
In quadrant II, what is the sign of sinθ?
Positive (y>0).
In quadrant II, sign of cosθ?
Negative (x<0).
In quadrant IV, sign of sinθ?
Negative.
In quadrant IV, sign of cosθ?
Positive.
At 225° (or 5π/4), is sinθ positive or negative?
Negative, quadrant III.
What is sin(210°) or sin(7π/6) on the unit circle?
−1/2.
What is cos(300°) or cos(5π/3) on the unit circle?
1/2.
If cosθ=√3/2 in quadrant I, which angle?
θ=30° or π/6.
If sinθ=−√2/2 in quadrant IV, which angle?
θ=315° or 7π/4.
If tanθ is positive, which quadrants does θ land in?
Quadrant I or III.
If θ=π/2, how many degrees is that?
90°.
If θ=2π/3, how many degrees is that?
120°.
If θ=4π/3, how many degrees is that?
240°.
If θ=5π/4, how many degrees is that?
225°.
θ=11π/6 is how many degrees?
330°.
