Trigonometric Functions and Values

Standard Angle Values

Trigonometric functions for specific angles are fundamental for solving problems in trigonometry. These values are often used in calculations and proofs.

• sin(37°) = \frac{3}{5}

• sin(53°) = \frac{4}{5}

• cos(37°) = \frac{4}{5}

• cos(53°) = \frac{3}{5}

• tan(37°) = \frac{3}{4}

• tan(53°) = \frac{4}{3}

These values are useful for quick reference and are derived from right triangle ratios.

Trigonometric Identities

Sum and Difference Formulas

Sum and difference formulas allow the calculation of trigonometric functions for the sum or difference of two angles.

• sin(A ± B) = sinA \cdot cosB ± cosA \cdot sinB

• cos(A ± B) = cosA \cdot cosB ∓ sinA \cdot sinB

• tan(A ± B) = \frac{tanA ± tanB}{1 ∓ tanA \cdot tanB}

Double Angle and Power-Reducing Formulas

Double angle formulas are used to express trigonometric functions of double angles in terms of single angles. Power-reducing formulas help simplify expressions involving squared trigonometric functions.

• cos(2θ) = cos^2θ - sin^2θ = 2cos^2θ - 1 = 1 - 2sin^2θ

• sin(2θ) = 2sinθ \cdot cosθ

Trigonometric Function Transformations

Co-function and Negative Angle Identities

Co-function identities relate trigonometric functions of complementary angles. Negative angle identities describe how trigonometric functions behave under sign changes.

• sin(90° - θ) = cosθ

• cos(90° - θ) = sinθ

• sin(-θ) = -sinθ

• cos(-θ) = cosθ

Graphs of Trigonometric Functions

Graphical Representation

Graphs of sine, cosine, and tangent functions illustrate their periodic nature and key properties such as amplitude, period, and phase shift.

• Sine and Cosine: Periodic with period

• Tangent: Periodic with period

• Key features: Maxima, minima, and points of intersection with the x-axis

Additional info:

Some images and formulas reference logarithmic and algebraic concepts, but the primary focus is on trigonometric identities, values, and graphs, which are directly relevant to college-level trigonometry.