Trigonometry

Section 8.1- Vocab Section

What type of triangle

if the c² value is less than a+b, it is an acute triangle. If the c squared is more than a+b, it is obtuse.

45-45-90 Triangle- Side lengths

if x^{2 +}x² = c² then it follows that c = x√2, where x is the length of each leg of the triangle. so, X times the square root of 2, giving the hypotenuse. In some situations, it is better to just round the square root instead of simplifying the radical, especially in real- world problems/ situations.

30-60-90 Triangle- Angles

• In a 30-60-90 triangle, the side lengths are in the ratio X: x√3: 2x. This means that if the length of the shorter leg (opposite the 30-degree angle) is x, then the longer leg (opposite the 60-degree angle) will be x√3, and the hypotenuse will be 2x.

• So, the smallest side is X, the hypotenuse is 2x, and the long leg is x√3, which correspond to angles of 30 degrees and 60 degrees, respectively.

  

Section 8.2- Trigonometric Ratios

There are SIX different trig. Ratios; but focus on 3. Cosine, Sine, and Tangent. All the trig. funcs. Can be written as ratios comparing side lengths.

Vocab

“ Theta” (θ) is used to represent a missing angle measure.

“ Ajacent” is the leg that touches θ

“ Opposite” is the leg not touching θ

Using Trig. Functions

sine(θ) = sin(θ) = opposite/ hypotenuse

cosine(θ) = cos(θ) = ajacent/ hypotnuse

tangent(θ) = tan(θ) = opposite/ajacent

Example Of Trig Function

Missing Side Measure

Missing Angle Measure

SOLVING REQUIRES INVERSE TRIG. FUNCTIONS (EXPONENT OF -1)

Section 8.3- The Law of Sines

Notes 8.3

sinA/a = sinB/b = sinC/c

Section 8.4- The Law of Cosines

Notes

c² = a² + b² − 2ab cos(C)

to remember: Pythagorean formula but add -2ab cos(C).

It’s only supposed to be used when you have SSS or SAS TO FIND THE 3RD SIDE/ANGLE— TO BE ABLE TO USE LAW OF SINES.

To find a missing angle, you solve it geometrically, when you get to cos(s); you use arcsin to find the result.

Section 8.5- Problem-Solving with Trigonometry

Vocab

Angle of elevation

• The angle is created by two rays, one parallel to the ground, and the second one above it, at the angle you're looking at.

Angle of depression

The angle created by two rays is the opposite of the angle of elevation.

use tangent for solving the problems most of the time

New area formula

1/2(b,c)sin(A)

Review

Important Formulas

Pythagorean fromula = a² +b² = c²

Triangle Type = if c²>a²+b² then it’s acute. if c²<a²+b² then its obtuse

454590 Triangle = x²+x²= c² then the hypotenuse is x√2

306090 Triangle = (1)x, √3(x), 2(x)

sin(θ)=opposite/hypotenuse, cos(θ) =adjacent -hypotenuse, tan(θ) = opposite/adjacent

Area formula: 1/2(b,c)sin(A)

Law of Sines: this second

Law of Cosines- this first

a² + b² − 2ab cos(C) = c²