Trigonometry Formulas

What is the formula for an area of a triangle?

A=(1/2)bh

Coterminal Angles

Angle in standard position with the same terminal side, but a different angle measure.

What are the steps for solving a triangle?

Determine how many side lengths you know. If you only know 1 side length, do you know 2 angle measures? If not, no solution. If you know 2, solve. Now, if you know two side lengths, do you know the measure of an angle not in between the two sides? If yes, is measure of <A ≥90°? If yes, is a>b? If a>b, you solve. If it is less than b, you have no triangle. But, if <A is less than 90°, is a<h? If yes, no triangle. If it it is greater than h, is it equal to h? If yes, solve. If not, is a≥b? If no, there are 2 triangles. If yes, solve.

Law of Cosines

For ▲ABC, the Law of Cosines states that: a²=b²+c²-2bccosA b²=a²+c²-2accosB c²=b²+c²-2abcosC

Law of Sines

For ▲ABC, the Law of Sines states that (SinA/a)=(SinB/b)= (SinC/c)

Heron's Formula

For ▲ABC, where s is half of the perimeter of the triangle, or (1/2)(a+b+c), Area=√s(s-a)(s-b)(s-c)

What is Cos^(-1)a=Θ

Inverse cosine with a restricted domain

What is Sin^(-1)a=Θ

Inverse sine with a restricted domain

What is Tan^(-1)a=Θ

Inverse tangent with a restricted domain

What does it mean when SinΘ, CosΘ, and TanΘ all start with a capital letter?

It means they have restricted domains

When can you use Law of Sines?

Only when 1. you know two angle measures and a side (AAS, ASA) 2. two sides and an angle measure not between them (SSA). SAS DOES NOT WORK!!

Reference Angle

Positive acute angle formed by the terminal side of Θ (angle measure) and the x-axis.

Arc Length Formula

S=rΘ

Standard Position

When the vertex is at the origin (0,0) and one ray is on the positive x-axis.

Ambiguous Case

Works for SSA

cosineΘ

adjacent / hypotenuse

What is the inverse of cosΘ= ?

cos^(-1)a=Θ

What are the Pythagorean Identities?

cos²Θ+sin²Θ=1 1+tan²Θ=sec²Θ cot²Θ+1=csc²Θ

cotangentΘ

cot; inverse of tangent; 1/tanΘ= adjacent / opposite

cosecantΘ

csc; inverse of sin; 1/sinΘ= hypotenuse / opposite

What are the reciprocal identities?

cscΘ=1/sinΘ secΘ=1/cosΘ cotΘ=1/tanΘ

tangentΘ

opposite / adjacent

sineΘ

opposite / hypotenuse

secantΘ

sec; inverse of cosine; 1/cosΘ= hypotenuse/adjacent

What are the Negative-Angle Identities?

sin(-Θ)=-sinΘ cos(-Θ)=cosΘ tan(-Θ)=-tanΘ

What are the Sum Identities?

sin(A+B)=sinAcosB+cosAsinB cos(A+B)=cosAcosB-sinAsinB tan(A+B)=(tanA+tanB) / (1-tanAtanB)

What are the Difference Identities?

sin(A-B)=sinAcosB-cosAsinB cos(A-B)=cosAcosB+sinAsinB tan(A-B)=(tanA-tanB) / (1+tanAtanB)

What are the Half-Angle Identities?

sin(Θ/2)= ±√(1-cosΘ) / 2 cos(Θ/2)=±√(1+cosΘ) / 2 tan(Θ/2)=±√(1-cosΘ) / (1+cosΘ)

What are the Double-Angle Identities?

sin2Θ=2sinΘcosΘ cos2Θ=cos²Θ-sin²Θ ← most common cos2Θ=2cos²Θ-1 cos2Θ=1-2sin²Θ tan2Θ= (2tanΘ) / (1-tan²Θ)

What is the inverse of sinΘ=a ?

sin^(-1)a=Θ

SOH CAH TOA CHO SHA COA

sine=o/h; cos=a/h; tan=o/a csc=h/o; sec=h/a; cot=a/o

What is the inverse of tanΘ=a ?

tan^(-1)a=Θ

What are the Tangent and Cotangent Ratio Identities?

tanΘ=sinΘ/cosΘ cotΘ=cosΘ/sinΘ tanΘ=secΘ/cscΘ

Degrees → Radians

times number of degrees by (πrad/180°)

Radians → Degrees

times number of radians by (180°/πrad)

Pythagorean Identity

x²+y²=r² → (x²/r²)+(y²/r²)=1 →cos²Θ+sin²Θ=1 (r=hypotenuse)

sinΘ (hand trick)

√bottom fingers/2

tanΘ (hand trick)

√bottom fingers/√top fingers

cosΘ (hand trick)

√top fingers/2