Trigonometry: Compound, Multiple, and Submultiple Angles

Flashcards covering compound angle identities, multiple angle formulas, transformation identities, specific trigonometric values, and periodic properties based on the lecture notes.

sin(A+B)

sin(A+B) = sinAcosB + cosAsinB

sin(A−B)

sin(A−B) = sinAcosB − cosAsinB

cos(A+B)

cos(A+B) = cosAcosB − sinAsinB

cos(A−B)

cos(A−B) = cosAcosB + sinAsinB

tan(A+B)

tan(A+B) = \frac{tanA + tanB}{1 − tanAtanB}

tan(A−B)

tan(A−B) = \frac{tanA − tanB}{1 + tanAtanB}

cot(A+B)

cot(A+B) = \frac{cotAcotB − 1}{cotA + cotB}

cot(A−B)

cot(A−B) = \frac{cotB − cotA}{cotAcotB + 1}

When can the tangent addition/subtraction formulas be used?

The relevant tangent values and denominator must be defined/non-zero.

sin(A+B)+sin(A−B)

2sinAcosB

sin(A+B)−sin(A−B)

2cosAsinB

cos(A+B)+cos(A−B)

2cosAcosB

cos(A−B)−cos(A+B)

2sinAsinB

sin(A+B)sin(A−B)

sin²A − sin²B

cos(A+B)cos(A−B)

cos²A − sin²B

tan(A+B)tan(A−B)

1 − tan²Atan²B / tan²A − tan²B

Find tan(45° + θ)

\frac{1 + tanθ}{1 − tanθ}

Find tan(45° − θ)

\frac{1 − tanθ}{1 + tanθ}

Find cot(45° + θ)

\frac{1 + tanθ}{1 − tanθ}

Find cot(45° − θ)

\frac{1 − tanθ}{1 + tanθ}

What is tan(45° + θ)tan(45° − θ)?

1

Express tan(45° + θ) using sine and cosine.

\frac{cosθ − sinθ}{cosθ + sinθ}

Express tan(45° − θ) using sine and cosine.

\frac{cosθ + sinθ}{cosθ − sinθ}

What is sin(A+B+C)?

sinAcosBcosC + cosAsinBcosC + cosAcosBsinC − sinAsinBsinC

What is cos(A+B+C)?

cosAcosBcosC − cosAsinBsinC − sinAcosBsinC − sinAsinBcosC

What is tan(A+B+C)?

\frac{tanA + tanB + tanC − tanAtanBtanC}{1 − tanAtanB − tanBtanC − tanCtanA}

What is cot(A+B+C)?

\frac{cotA + cotB + cotC − cotAcotBcotC}{1 − cotAcotB − cotBcotC − cotCcotA}

If A+B+C=nπ, what relation holds between the tangents?

tanA + tanB + tanC = tanAtanBtanC

If A+B+C=nπ, what cotangent identity holds?

cotAcotB + cotBcotC + cotCcotA = 1

If A+B+C=2(2n+1)π, what tangent identity holds?

tanAtanB + tanBtanC + tanCtanA = 1

If A+B+C=2(2n+1)π, what cotangent identity holds?

cotA + cotB + cotC = cotAcotBcotC

What are multiple angles?

Angles such as 2A, 3A, 4A,…

What are sub-multiple angles?

Angles such as \frac{A}{2}, \frac{A}{3}, \frac{A}{4},…

What is sin(2A)?

2sinAcosA

What is cos(2A)?

cos²A − sin²A

Give two alternate forms of cos(2A).

2cos²A − 1, 1 − 2sin²A

What is tan(2A)?

\frac{2tanA}{1 − tan²A}

What is cot(2A)?

\frac{2cotA}{cot²A − 1}

Express sin(2A) in terms of tanA.

\frac{2tanA}{1 + tan²A}

Express cos(2A) in terms of tanA.

\frac{1 - tan²A}{1 + tan²A}

What is sin(3A)?

3sinA − 4sin³A

What is cos(3A)?

4cos³A − 3cosA

What is tan(3A)?

\frac{3tanA − tan³A}{1 − 3tan²A}

What is cot(3A)?

\frac{3cotA − cot³A}{1 − 3cot²A}

What is sin²A?

±\frac{1 - cosA}{2}

What is cos²A?

±\frac{1 + cosA}{2}

What is tan²A?

±\frac{1 + cosA}{1 - cosA}

What is another form of tan²A?

\frac{sinA}{1 - cosA}

Another form of tan²A?

\frac{1 + cosA}{sinA}

What determines the sign in half-angle formulas?

The quadrant in which A/2 lies.

What is cot²A?

\frac{2sin(A/2)cos(A/2)}{cos²(A/2) − sin²(A/2)}

What is sinA in terms of tan(A/2)?

sinA = \frac{2tan(A/2)}{1 + tan²(A/2)}

What is cosA in terms of tan(A/2)?

cosA = \frac{1 - tan²(A/2)}{1 + tan²(A/2)}

What is tanA in terms of tan(A/2)?

tanA = \frac{2tan(A/2)}{1 - tan²(A/2)}

What is sin(15°)?

\frac{2 - \sqrt{3}}{2}

What is cos(15°)?

\frac{\sqrt{6} + \sqrt{2}}{4}

What is tan(15°)?

2 - \sqrt{3}

What is cot(15°)?

2 + \sqrt{3}

What is sec(15°)?

\frac{4}{\sqrt{6} - \sqrt{2}}

What is cosec(15°)?

\frac{4}{\sqrt{6} + \sqrt{2}}

What is sin(18°)?

\frac{\sqrt{5} - 1}{4}

What is tan(18°)?

\sqrt{5 - 2\sqrt{5}}

What is tan(36°)?

\sqrt{5 + 2\sqrt{5}}

What is sin(54°)?

\frac{\sqrt{5} + 1}{4}

What is sin(72°)?

\frac{\sqrt{10 + 2\sqrt{5}}}{4}

What is sin(81°)?

\frac{\sqrt{2} + \sqrt{6}}{4} + \frac{\sqrt{5}}{4}

What is sin(87°)?

\frac{\sqrt{6} + 2}{4} + \frac{\sqrt{1 + \sqrt{3}}}{2}

What is tan(82.5°)?

\frac{3 + 2\sqrt{3}}{3 - 2\sqrt{3}}

What is tan(72°)?

\frac{4}{\sqrt{5} - 1}

What is tan(67.5°)?

\frac{1 + \sqrt{3}}{2}

What is sinC+sinD?

2sin\left(\frac{C+D}{2}\right)cos\left(\frac{C−D}{2}\right)

What is sinC−sinD?

2cos\left(\frac{C+D}{2}\right)sin\left(\frac{C−D}{2}\right)

What is cosC+cosD?

2cos\left(\frac{C+D}{2}\right)cos\left(\frac{C−D}{2}\right)

What is cosC−cosD?

−2sin\left(\frac{C+D}{2}\right)sin\left(\frac{C−D}{2}\right)

If cosA + cosB = a and sinA + sinB = b, find tan(2A + 2B).

tan(2A + 2B) = \frac{ab}{a^{2} + b^{2}}.

Under the same conditions, find sin(A + B).

sin(A + B) = \frac{a^{2} + b^{2}}{2ab}.

Under the same conditions, find cos(A + B).

cos(A + B) = \frac{a^{2} - b^{2}}{a^{2} + b^{2}}.

Under the same conditions, find tan(A + B).

tan(A + B) = \frac{a^{2} - b^{2}}{2ab}.

What is sinθ+sin(120°−θ)+sin(120°+θ)?

\frac{3}{2}.

What is cosθ+cos(120°−θ)+cos(120°+θ)?

0.

What is sinθsin(60°−θ)sin(60°+θ)?

\frac{1}{4}sin3θ.

What is cosθcos(60°−θ)cos(60°+θ)?

\frac{1}{4}cos3θ.

If α = 60°, 120°, 240°, 300°, what special tangent sum appears?

tanθ + tan(α + θ) + tan(α − θ) = 3tan³θ.

What is the corresponding cotangent result?

cotθ + cot(α − θ) + cot(α + θ) = 3cot³θ.

What is sin²θ + cos²θ?

1.

What is sin⁴θ + cos⁴θ?

1 − \frac{1}{2}sin²2θ.

What is sin⁶θ + cos⁶θ?

1 − \frac{3}{4}sin²2θ.

What is cotA + tanA?

2cosec²A.

What is cotA − tanA?

2cot²A.

What is tan(4π + A) + tan(4π − A)?

2sec²A.

What is (1 + secθ)(1 + sec²θ)⋯(1 + sec²nθ)?

tanθtan(2nθ).

What is cosθcos2θcos4θ⋯cos2n−1θ?

2^{n}sinθsin2nθ.

What is the general sine sum?

sinα + sin(α + β) + ⋯ + sin[α + (n − 1)β] = \frac{1}{2}sin\left(\frac{nβ}{2}\right)sin\left(α + \frac{(n−1)β}{2}\right).

What is the general cosine sum?

cosα + cos(α + β) + ⋯ + cos[α + (n − 1)β] = \frac{1}{2}sin\left(\frac{nβ}{2}\right)cos\left(α + \frac{(n−1)β}{2}\right).

Definition of periodic function

f(x + p) = f(x) for a positive p. The least positive such p is the fundamental period.

Period of sinx, cosx

2π.

Period of tanx, cotx

π.

Period of secx, cosecx

2π.

Period of f(ax + b)

|a|p if f(x) has period p.

Period of sin(ax) + cos(bx)

gcd(|a|, |b|)2π.