Chapter 9 - Introduction to Trigonometry Flashcards by Shahd Khoury

How do you find out all the formulas for sine, cosine & tangent? (4)

SOH CAH TOA
Sine: Opposite/Hypotenuse
Cosine: Adjacent/Hypotenuse
Tangent: Opposite/Adjacent

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What is everything to know about angle of rotations? (8)
(Angle of Rotation, Initial Side, Terminal Side, Standard Position, +/- Rotation, Reference Angle, Coterminal Angle)

AoR: Angle formed by rotating ray around vertex at origin thru four quadrants
Standard Position: Angle w/ vertex at origin & initial side on +x‑axis
Initial Side: Ray on +x‑axis, Terminal Side: Ray’s Final Position after Rotation
- Rotation is CC & - Rotation is C
Reference Angle: Positive acute angle b/w terminal side & x‑axis
Coterminal Angles: Angles that share the same terminal side

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Definition of Angle of Elevation & Angle of Depression (3)

Elevation: Angle measured upward from horizontal line of sight (lower acute angle of triangle)
Depression: Angle measured downward from horizontal line of sight (angle next to the upper acute angle of right triangle)
Angle of Depression = Angle of Elevation

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What are coterminal angles, and how do you find the smallest positive coterminal angle?

Def: Share same terminal side but differ in measure (sum up to multiple of 360)
To find the smallest positive coterminal angle:
–> If given angle is positive, subtract 360° until it lies b/w 0° & 360°
–> If given angle is negative, add 360° until it lies b/w 0° & 360°

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Definition of Reference Angle (3)

Every non‑quadrantal angle has reference angle, which helps evaluate sin, cos, & tan
Positive acute angle b/w terminal side of angle & x‑axis
Easy way to remember:“How far the angle is from the x‑axis”

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What are the trigonometric reciprocals? (5)

csc θ (cosecant) is the reciprocal of sin θ → H/O
2, sec θ (secant) is the reciprocal of cos θ → H/A
cot θ is the reciprocal of tan θ → A/O
If confused in finding them, just find its reciprocal & flip fraction
Their signs follow their reciprocals in ASTC pattern

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What is the 30, 45, 60 table for sin/cos/tan and their reciprocals?

Angles: 30° 45° 60°
sin: 1/2 √2/2 √3/2
cos: √3/2 √2/2 1/2
tan: √3/3 1 √3
csc: 2 √2 2√3/3
sec: 2√3/3 √2 2
cot: √3 1 √3/3

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How do you memorize the table for 30, 45, 60 of sin/cos/tan and their reciprocals?

Sin: 1, 2, 3, all radicals, then divide by 2
Cos: 3, 2, 1, all radicals, then divide by 2
Tan: 3, 1, 3, all radicals, first is over 3
Csc: Reciprocal of Sin
Sec: Reciprocal of Cos
Cot: Reciprocal of Tan

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How do you find the value of a trigonometric function when you’re given an angle? (4)
Ex: Cos210

Find reference angle by sketching quadrants & measuring positive acute angle b/w terminal side & x‑axis
Use ASTC rule to determine whether trig value is +/- based on quadrant (applies to reciprocals too)
If the angle is 30°/45°/60°, use special‑angles table to get exact value
If it’s not a special angle:
- Draw right triangle using given trig information (H/O/A) (usually given diff trig function)
- Use the Pythagorean theorem to find the missing side (will be negative depending on quadrant)
- Plug side lengths into trig ratio you need

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What is the ASTC chart? (5)

Quadrant 1: All Trig Functions are Positive
Quadrant 2: Sine & Cosecant is Positive
Quadrant 3: Tangent & Cotangent is Positive
Quadrant 4: Cosine & Secant is Positive
Mnemonic: All Students Take Classes

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What is the main thing to remember about a reference angle? (2)

Reference angle tells you how far an angle is from the x axis
If given quadrant & reference angle, can determine value of angle

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If you are given a trig equation of the form ___θ = ___ and a sign condition like ___θ > 0, how do you find θ? (3)
Ex: If sinθ = -1/2 & tanθ > 0, find θ

Find Quadrant Location (ASTC) to see where trig function is +/-
Ex: Tangent is positive in Quadrants I and III, but Sine is negative, so it’s QIII
Find Reference Angle from Given Trig Function: Use Special Angles Table
Ex: sinθ = -1/2 –> 30 Degrees
Combine quadrant & reference angle to determine the actual angle (θ)
Ex: 180 + 30 = 210 Degrees

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When solving for the value of a trigonometric function & the reference angle is NOT in the special angles table, how do you solve? (3)
Ex: If cosθ = -2/3 & sinθ > 0, what’s tanθ?

Draw right triangle using given trig information (H/O/A) (usually given diff trig function)
Ex: cosθ = A/H, cosθ = -2/3, adjacent = -2, hypotenuse = 3
Use the Pythagorean theorem to find the missing side
Ex: opp² + (-2)² = 3² –> opposite = √5
Plug side lengths into trig ratio you need
tanθ = O/A, √5 / -2 –> tanθ = -√5/2

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What do you do if they give you an angle that terminates at a certain coordinate and must find the exact value? (4)

Determine Quadrant Location
Draw a Triangle in Quadrant w/ side length = x, side length = y
Use Pythagorean theorem to find third side
Use SOHCAHTOA to find Trig Function

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What do you do if they give you an angle that terminates at a certain coordinate and must find the exact value of a trigonometric function? (2)

Given Coordinate (X, Y)
1. Sin = Y, Cos = X, Tan = Y/X
2. Reciprocals (Csc/Sec/Cot) is reciprocal of those fraction

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Definition of Radian (2)

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Measure of a central angle that intercepts an arc whose length = radius
Formula: s = θr (arc length = angle in radians/length of radius)

What are the common angle measures in radian form?
(360, 270, 180, 90, 60, 45, 30)

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2π = 360
3π/2 = 270
π = 180
π/2 = 90
π/3 = 60
π/4 = 45
π/6 = 30

How do you convert radians to degrees?
How do you convert degrees to radians?

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Radians to Degrees: Multiply by 180/π
Degrees to Radians: Multiply by π/180

How do you find the measure to tan/sin/cos (radian)? (3)

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Stay in radians, don’t convert to degrees & work w/ π‑based angles
Identify reference angle using special angles (π/6, π/4, π/3) & use chart
Draw coordinate plane:
- π on left of x - axis and 2π on right of x acis
- Locate Angle on Quadrant
- Use Quadrant Location (ASTC) to determine if +/-

What is the unit circle?

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Circle centered at origin (0,0) w/ radius of 1
Four key coordinates:
- Right: 0° / 360° → (1, 0)
- Up: 90° → (0, 1)
- Left: 180° → (-1, 0)
- Down: 270° → (0, -1)
Used to determine sin/cos/tan/csc/sec/cot at 0°/90°/180°/270°/360°
(cos θ, sin θ) = (x, y)

How do you find the exact value of sin/cos/tan/csc/sec/cot AT 0°, 90°, 180°, 270°, 360°

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Use the Unit Circle:
- 0° & 360° → (1, 0)
- 90° → (0, 1)
- 180° → (−1, 0)
- 270° → (0, −1)
From each point:
cos(θ) = x‑coordinate
sin(θ) = y‑coordinate
tan(θ) = sin(θ) / cos(θ)
For reciprocals:
csc(θ) = 1 / sin(θ)
sec(θ) = 1 / cos(θ)
cot(θ) = 1 / tan(θ)

How do you determine measure of central angle in radians of a circle?
How do you determine the area of a sector in radians of a circle?

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s = θr (arc length = angle in radians/length of radius)
A = 1/2 θr²

Definition of Law of Sines (3/1)
(Use When, Formula)

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Use when:
- NOT a right triangle
- Missing side/angle
- Given SAS/AAS
Formula: a/sinA = b/sinB

Definition of Law of Cosines (Used When, Formula, Note, Note)

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Used when:
- NOT right triangle
- Missing side/angle
- Given SSS/SAS
Formula: a² = b² + c² − 2bc·cos(A)
Note: Squared side opposite = sign MUST match angle in cosine
Note: “−2bc” is attached to cos(A) & must divide when solving, do NOT combine w/ b² + c²

How do you find the area of a non-right triangle w/o height? (3)

1. If given SAS: ½ab·sin(C) 2. If given SSS: Heron's Formula 3. If given AAS: Law of Sines (find missing side), ½ab·sin(C)

What is Heron's formula? (2)

1. Use when: Given SSS 2. Formula: s = (a + b + c) / 2 & A = √[s(s−a)(s−b)(s−c)]

How do you solve forces word problems (5)

1. Force 1 & Force 2 are rays in a vector diagram 2. Resultant force is dotted line b/w two rays 3. Complete diagram as parallelogram & use opposite‑angles/opposite‑sides (and Z‑pattern angle relationships) to identify equal or supplementary angles 4. This creates two triangles (choose one) 5. Use the Law of Cosines/Law of Sines to solve for missing side/angle

How do you know which force is larger in a force word problem?

1. Larger angle is always opposite larger side in a triangle 2. In a parallelogram, opposite sides are equal, so each force corresponds to one side of the vector triangle 3. Since each force is a side of the vector diagram, the force opposite the larger angle is the larger force

How do you solve ambiguous cases (number of triangles that can be formed?) (5)

1. Given: First angle, Know only 0/1/2 triangles form 2. Use Law of Sines to find measure of missing angle opposite known side 2. Round angle to the nearest whole number, then find the third angle (since all angles must add to 180°) 3. If the three angles add to 180°, then at least one triangle exists 4. To check for a second triangle: Keep given angle same, replace second angle w/ supplement (180° - angle) 5. If new set of angles still add to 180°, then second triangle can form 6. Note: When doing Law of Sines,

What are some things you need to know when sketching a right triangle in the quadrants to find the exact value of a trig function? (3)

- Hypotenuse must hit center of coordinate plane - Hypotenuse is always positive - Only legs) can be positive/negative depending on quadrant

Chapter 9 - Introduction to Trigonometry Flashcards

(30 cards)