Math ch 6

Identities of trigonometry
Here are the proofs for the trigonometric identities:

*Fundamental Identities*

1. Pythagorean Identity: sin^2(x) + cos^2(x) = 1

Proof:

Consider a right-angled triangle with sides of length 1, x, and hypotenuse.

sin(x) = opposite/hypotenuse = x/1 = x

cos(x) = adjacent/hypotenuse = 1/1 = 1

By Pythagoras' theorem:

x^2 + 1^2 = 1^2

x^2 + 1 = 1

sin^2(x) + cos^2(x) = x^2 + 1 = 1

1. Sum of Squares Identity: tan^2(x) + 1 = sec^2(x)

Proof:

tan(x) = sin(x)/cos(x)
sec(x) = 1/cos(x)

tan^2(x) + 1 = (sin^2(x)/cos^2(x)) + 1

= (sin^2(x) + cos^2(x))/cos^2(x)

= 1/cos^2(x)

= sec^2(x)

1. Difference of Squares Identity: 1 + cot^2(x) = csc^2(x)

Proof:

cot(x) = cos(x)/sin(x)

csc(x) = 1/sin(x)

1 + cot^2(x) = 1 + (cos^2(x)/sin^2(x))

= (sin^2(x) + cos^2(x))/sin^2(x)

= 1/sin^2(x)

= csc^2(x)

*Angle Addition and Subtraction Identities*

1. Sin(a + b) = sin(a)cos(b) + cos(a)sin(b)

Proof:

Consider two right-angled triangles with angles a and b.

sin(a + b) = opposite/hypotenuse

= (sin(a)cos(b) + cos(a)sin(b))/1

1. Cos(a + b) = cos(a)cos(b) - sin(a)sin(b)

Proof:

cos(a + b) = adjacent/hypotenuse

= (cos(a)cos(b) - sin(a)sin(b))/1

1. Tan(a + b) = (tan(a) + tan(b)) / (1 - tan(a)tan(b))

Proof:

tan(a + b) = sin(a + b)/cos(a + b)

= (sin(a)cos(b) + cos(a)sin(b))/(cos(a)cos(b) - sin(a)sin(b))

= (tan(a) + tan(b)) / (1 - tan(a)tan(b))

*Double Angle Identities*

1. Sin(2x) = 2sin(x)cos(x)
Proof:

sin(2x) = 2sin(x)cos(x)

= 2(opposite/hypotenuse)(adjacent/hypotenuse)

1. Cos(2x) = cos^2(x) - sin^2(x)

Proof:

cos(2x) = cos^2(x) - sin^2(x)

= (adjacent/hypotenuse)^2 - (opposite/hypotenuse)^2

1. Tan(2x) = 2tan(x) / (1 - tan^2(x))

Proof:

tan(2x) = sin(2x)/cos(2x)

= (2sin(x)cos(x))/(cos^2(x) - sin^2(x))

= 2tan(x) / (1 - tan^2(x))

*Half Angle Identities*

1. Sin(x/2) = ±√((1 - cos(x))/2)

Proof:

sin(x/2) = √((1 - cos(x))/2)

= √(1 - (adjacent/hypotenuse)^2)/2

1. Cos(x/2) = ±√((1 + cos(x))/2)

Proof:

cos(x/2) = √((1 + cos(x))/2)

= √(1 + (adjacent/hypotenuse)^2)/2

1. Tan(x/2) = (1 - cos(x)) / sin(x)

Proof:

tan(x/2) = sin(x/2)/cos(x/2)

= (1 - cos(x))/sin(x)

*Product-to-Sum Identities*

1. Sin(a)cos(b) = (1/2)[cos(a-b) + cos(a+b)]

Proof:

sin(a)cos(b) = (1/2)[cos(a-b) + cos(a+b)]

= (1/2)[(adjacent/hypotenuse) + (adjacent/hypotenuse)]

Here are the rest of the proofs:

_Product-to-Sum Identities (continued)_

1. Cos(a)cos(b) = (1/2)[cos(a-b) + cos(a+b)]

Proof:

cos(a)cos(b) = (1/2)[cos(a-b) + cos(a+b)]

= (1/2)[(adjacent/hypotenuse) + (adjacent/hypotenuse)]

1. Sin(a)sin(b) = (1/2)[cos(a-b) - cos(a+b)]

Proof:

sin(a)sin(b) = (1/2)[cos(a-b) - cos(a+b)]

= (1/2)[(opposite/hypotenuse) - (opposite/hypotenuse)]

_Sum-to-Product Identities_

1. Sin(a) + Sin(b) = 2sin((a+b)/2)cos((a-b)/2)

Proof:

sin(a) + sin(b) = 2sin((a+b)/2)cos((a-b)/2)

= 2(opposite/hypotenuse)(adjacent/hypotenuse)

1. Cos(a) + Cos(b) = 2cos((a+b)/2)cos((a-b)/2)

Proof:

cos(a) + cos(b) = 2cos((a+b)/2)cos((a-b)/2)

= 2(adjacent/hypotenuse)(adjacent/hypotenuse)

1. Sin(a) - Sin(b) = 2cos((a+b)/2)sin((a-b)/2)

Proof:

sin(a) - sin(b) = 2cos((a+b)/2)sin((a-b)/2)

= 2(opposite/hypotenuse)(opposite/hypotenuse)

Formala :
Here are some trigonometry formulas for specific triangles:

*Right Triangle*

1. Pythagorean Theorem: a^2 + b^2 = c^2 (where c is the hypotenuse)

1. Sine: sin(A) = opposite/hypotenuse = a/c

1. Cosine: cos(A) = adjacent/hypotenuse = b/c

1. Tangent: tan(A) = opposite/adjacent = a/b

*Oblique Triangle*

1. Law of Sines: a/sin(A) = b/sin(B) = c/sin(C)

1. Law of Cosines: c^2 = a^2 + b^2 - 2ab*cos(C)

*Equilateral Triangle*

1. All angles are 60°

1. All sides are equal

*Isosceles Triangle*

1. Two sides are equal

1. Two angles are equal

*30-60-90 Triangle*

1. Angles: 30°, 60°, 90°

1. Side ratios: 1:√3:2

*45-45-90 Triangle*

1. Angles: 45°, 45°, 90°

1. Side ratios: 1:1:√2

* Trigonometry Ratio *

1. If sec(x) = 5/3, find tan(x) and sin(x).

Solution:

sec(x) = 5/3
tan(x) = √(sec^2(x) - 1) = √((5/3)^2 - 1) = 4/3
sin(x) = 1/sec(x) = 3/5

1. If cosec(x) = 7/6, find cot(x) and cos(x).

Solution:

cosec(x) = 7/6
cot(x) = √(cosec^2(x) - 1) = √((7/6)^2 - 1) = 6/√7
cos(x) = 1/cosec(x) = 6/7

*Identities and Formulas*

1. Prove: tan(A - B) = (tan(A) - tan(B)) / (1 + tan(A)tan(B))

Solution:

Start with the identity: tan(A - B) = (sin(A)cos(B) - cos(A)sin(B)) / (cos(A)cos(B)

+ sin(A)sin(B))
Divide numerator and denominator by cos(A)cos(B)
Use the definitions of tan(A) and tan(B)

1. Simplify: 2cos^2(x) - 1

Solution:

Use the identity: cos(2x) = 2cos^2(x) - 1

So, 2cos^2(x) - 1 = cos(2x)

*Graphs of Trigonometric Functions*

1. Sketch the graph of y = sin(2x) - 1.

Solution:

Amplitude = 1
Period = π
Phase shift = 0
Vertical shift = -1

1. Find the amplitude and period of y = 2sin(x/2).

Solution:

Amplitude = 2
Period = 4π

*Applications*

1. A Ferris wheel with radius 10 m rotates at 5 rpm. Find the linear speed.
Solution:

Angular speed = 5 rpm = 5(2π)/60 rad/s

Linear speed = radius × angular speed = 10 × 5(2π)/60 = 5.24 m/s

1. A surveyor measures the angle of elevation to the top of a building as 60°. If

the distance to the building is 50 m, find the height.

Solution:

Use the tangent function: tan(60°) = height / 50

Height = 50 × tan(60°) = 50√3 m

*Challenge Questions*

1. Prove: sin(5x) = 16sin^5(x) - 20sin^3(x) + 5sin(x)

Solution:

Use De Moivre's theorem or multiple angle formulas

1. Find the value of cos(15°).

Solution:

Use the identity: cos(15°) = cos(45° - 30°)

= cos(45°)cos(30°) + sin(45°)sin(30°)
= (√2/2)(√3/2) + (√2/2)(1/2)
= (√3 + 1)/2√2

Here are some examples and practice problems for trigonometry:

*Examples*

1. Find sin(x) if cos(x) = 3/5.

Solution: sin(x) = √(1 - cos^2(x)) = √(1 - (3/5)^2) = 4/5

1. Find tan(x) if sin(x) = 2/3.

Solution: tan(x) = sin(x)/cos(x) = (2/3)/√(1 - sin^2(x)) = 2/√5

1. Find cos(2x) if cos(x) = 2/3.

Solution: cos(2x) = cos^2(x) - sin^2(x) = (2/3)^2 - (1 - (2/3)^2) = 1/3

*Practice Problems*

*Basic Trigonometry*

1. Find sin(x) if cos(x) = 2/5.

1. Find tan(x) if sin(x) = 3/4.

1. Find cos(x) if tan(x) = 5/12.

*Trigonometric Identities*

1. Prove: sin(a+b) = sin(a)cos(b) + cos(a)sin(b)

1. Simplify: cos^2(x) + sin^2(x)

1. Prove: tan(2x) = 2tan(x)/(1 - tan^2(x))

*Solving Triangles*

1. In a right triangle, find the length of the hypotenuse if one leg is 3 and the other
leg is 4.

1. In a right triangle, find the measure of an angle if the opposite side is 5 and the
hypotenuse is 13.

1. In an oblique triangle, find the measure of an angle if two sides are 5 and 7.

*Word Problems*

1. A ship sails 20 miles east and then 30 miles north. Find the distance and
direction from the starting point.

1. A plane flies 200 miles at an angle of 30°. Find the horizontal and vertical
components.

1. A force of 50 N acts at an angle of 45°. Find the x and y components.

Last modified: Sep 26, 2024