09/10 Trigonometric Equations and Applications Free Online FlashCards

What is a trigonometric identity?

A trigonometric identity is an equation true for every value of the variable for which both sides are defined.

State the three reciprocal trigonometric identities.

The reciprocal identities are csc θ = 1/sin θ, sec θ = 1/cos θ, and cot θ = 1/tan θ.

What is the primary Pythagorean identity?

The primary Pythagorean identity is sin²θ + cos²θ = 1.

State the sine sum identity.

sin(α + β) = sin α cos β + cos α sin β.

How should a trigonometric identity be verified?

Transform one side into the other through a connected sequence of equivalent expressions, usually starting with the more complicated side.

Give the general solutions to sin θ = k.

If α = sin⁻¹(k), then θ = α + 2πn or θ = π − α + 2πn, where n ∈ ℤ.

Solve 2 cos x + 1 = 0 on [0, 2π).

On [0, 2π), 2 cos x + 1 = 0 gives cos x = −1/2, so x = 2π/3 and 4π/3.

Give the general solutions to tan θ = k.

If α = tan⁻¹(k), then θ = α + πn, where n ∈ ℤ, because tangent has period π.

What is the principal-value range of cos⁻¹x?

The range of cos⁻¹x is [0, π]. Its principal angle lies in Quadrant I or II.

What is the difference between sin⁻¹x and 1/sin x?

sin⁻¹x means inverse sine, whereas 1/sin x is the reciprocal csc x.

Solve 2 sin²x − sin x − 1 = 0 on [0, 2π).

Factoring gives sin x = −1/2 or sin x = 1; thus x = π/2, 7π/6, and 11π/6 on [0, 2π).

When is the Law of Sines useful?

Use the Law of Sines for ASA, AAS, and SSA information. The SSA case can produce zero, one, or two triangles.