Trigonometry Formulas and Applications

10 Questions

What is the Pythagorean Identity, and how is it used to verify trigonometric identities?

The Pythagorean Identity is sin²θ + cos²θ = 1. It is used to verify trigonometric identities by substituting sin²θ or cos²θ into the identity to simplify expressions.

Derive the sum formula for cosine, cos(a + b) = cos(a)cos(b) - sin(a)sin(b), using the unit circle.

Starting with the unit circle, we can derive the formula by using the x- and y-coordinates of points on the circle.

Prove that tan(2θ) = (2tan(θ)) / (1 - tan²(θ)) using the double angle formula for sine and cosine.

Start with the double angle formulas for sine and cosine, then divide the sine formula by the cosine formula to obtain the tangent formula.

What is the cotangent of an angle, and how is it related to the sine and cosine ratios?

The cotangent of an angle is the ratio of the adjacent side to the opposite side. It is the reciprocal of the tangent ratio.

Solve the right triangle with legs 3 and 4, and find the length of the hypotenuse using the Pythagorean theorem.

The length of the hypotenuse is √(3² + 4²) = 5.

In an oblique triangle, if a = 5, b = 7, and C = 60°, find the length of side c using the law of cosines.

c² = 5² + 7² - 2(5)(7)cos(60°), so c ≈ 8.60.

How is trigonometry used in medical imaging to reconstruct 3D models from 2D images?

Trigonometry is used to apply transformations to 2D images, allowing for the reconstruction of 3D models.

What is the law of sines, and how is it used to solve triangles?

The law of sines states that a / sin(A) = b / sin(B) = c / sin(C). It is used to find missing sides or angles in triangles.

Derive the double angle formula for sine, sin(2θ) = 2sin(θ)cos(θ), using the sum formula for sine.

Start with the sum formula for sine, then substitute θ for b and use the Pythagorean Identity to simplify.

How is trigonometry used in navigation to determine distances and directions between locations?

Trigonometry is used to calculate distances and directions using triangulation methods.

Study Notes

Identities and Formulas

Pythagorean Identity: sin²θ + cos²θ = 1
Sum and Difference Formulas:
- sin(a + b) = sin(a)cos(b) + cos(a)sin(b)
- cos(a + b) = cos(a)cos(b) - sin(a)sin(b)
- tan(a + b) = (tan(a) + tan(b)) / (1 - tan(a)tan(b))
Double and Half Angle Formulas:
- sin(2θ) = 2sin(θ)cos(θ)
- cos(2θ) = cos²(θ) - sin²(θ)
- tan(2θ) = (2tan(θ)) / (1 - tan²(θ))
Product-to-Sum Formulas:
- sin(a)cos(b) = (1/2)[sin(a + b) + sin(a - b)]
- cos(a)sin(b) = (1/2)[sin(a + b) - sin(a - b)]

Trigonometric Ratios

Sine (sin): opposite side / hypotenuse
Cosine (cos): adjacent side / hypotenuse
Tangent (tan): opposite side / adjacent side
Cotangent (cot): adjacent side / opposite side
Secant (sec): hypotenuse / adjacent side
Cosecant (csc): hypotenuse / opposite side

Solving Triangles

Right Triangles:
- Use trigonometric ratios to find missing sides or angles
- Apply Pythagorean theorem to find hypotenuse
Oblique Triangles:
- Use law of sines or law of cosines to find missing sides or angles
- Law of sines: a / sin(A) = b / sin(B) = c / sin(C)
- Law of cosines: c² = a² + b² - 2ab * cos(C)

Applications of Trigonometry

Geometry and Physics:
- Calculate distances, heights, and angles in triangles
- Model periodic phenomena (e.g., sound waves, light waves)
Navigation and Surveying:
- Determine distances and directions between locations
- Calculate heights and angles of landmarks
Medical Imaging and Computer Graphics:
- Use trigonometry to reconstruct 3D models from 2D images
- Apply trigonometric transformations to manipulate objects in 3D space