Trig Identities

Summary

This video covers essential trigonometric identities, formulas, and concepts typically encountered in a trigonometry course. It starts with fundamental ratios like sine, cosine, and tangent, explaining their relationships through the acronym SOHCAHTOA. The video also explores reciprocal identities, quotient identities, Pythagorean identities, and the difference between even and odd functions. Additionally, it dives into cofunction, double-angle, half-angle, and sum/difference identities. Practical examples and a breakdown of key formulas such as the Law of Sines, Law of Cosines, and area calculation methods are included. Ideal for students starting trigonometry or preparing for exams.

Takeaways

• 😀 Trigonometric ratios are based on the SOHCAHTOA acronym, representing sine, cosine, and tangent functions using opposite, adjacent, and hypotenuse sides of a right triangle.
• 😀 Reciprocal identities involve sine, cosine, and tangent, where cosecant, secant, and cotangent are the reciprocals of sine, cosine, and tangent respectively.
• 😀 The three key Pythagorean identities you need to know are: sine^2 + cosine^2 = 1, 1 + cotangent^2 = cosecant^2, and 1 + tangent^2 = secant^2.
• 😀 Odd and even trigonometric functions: Sine, tangent, and cosecant are odd functions, while cosine and secant are even functions.
• 😀 Cofunction identities describe the relationships between sine and cosine, secant and cosecant, and tangent and cotangent at complementary angles (adding to 90 degrees).
• 😀 Double angle identities are important for simplifying expressions with doubled angles, such as sine(2θ) = 2sin(θ)cos(θ) and cosine(2θ) = cos²(θ) - sin²(θ).
• 😀 Half angle identities provide useful formulas for calculating trigonometric values for half of an angle, such as sine(θ/2) = ±√(1 - cos(θ))/2.
• 😀 Sum and difference identities help in simplifying expressions involving sums or differences of angles, such as sin(α ± β) = sin(α)cos(β) ± cos(α)sin(β).
• 😀 Power-reducing formulas help reduce the powers of trigonometric functions, such as sin²(θ) = (1 - cos(2θ))/2 and cos²(θ) = (1 + cos(2θ))/2.
• 😀 The Law of Sines and Law of Cosines are crucial for solving triangles, allowing the calculation of unknown sides or angles in non-right triangles.

Q & A

What is the key mnemonic to remember the basic trigonometric ratios?

The key mnemonic is 'SOHCAHTOA', which stands for Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, and Tangent = Opposite/Adjacent.

What are the reciprocal identities in trigonometry?

The reciprocal identities are: Cosecant (csc) = 1/Sine, Secant (sec) = 1/Cosine, and Cotangent (cot) = 1/Tangent.

How do you calculate the six trigonometric ratios for a right triangle with sides 3, 4, and 5?

For a triangle with sides 3, 4, and 5, the ratios are: Sine = 4/5, Cosine = 3/5, Tangent = 4/3, Cosecant = 5/4, Secant = 5/3, and Cotangent = 3/4.

What is the Pythagorean identity in trigonometry?

The Pythagorean identity is: Sine²(θ) + Cosine²(θ) = 1.

How are the quotient identities derived from the basic trigonometric ratios?

The quotient identities are derived as follows: Tangent (tan) = Sine/Cosine, and Cotangent (cot) = Cosine/Sine.

What is the difference between even and odd trigonometric functions?

Even trigonometric functions (like Cosine and Secant) remain unchanged when the angle is negated, while odd trigonometric functions (like Sine, Tangent, Cosecant, and Cotangent) change sign when the angle is negated.

What are cofunction identities, and can you provide an example?

Cofunction identities relate pairs of trigonometric functions. For example, Cosine(θ) = Sine(90° - θ), and vice versa. Similarly, Secant(θ) = Cosecant(90° - θ).

What is the formula for the double angle identity of sine?

The double angle identity for sine is: Sine(2θ) = 2 Sine(θ) Cosine(θ).

What is the half-angle identity for cosine?

The half-angle identity for cosine is: Cosine(θ/2) = ±√(1 + Cosine(θ))/2.

What is the Law of Sines and how is it used?

The Law of Sines states that the ratio of the sine of an angle to the length of the opposite side is constant for all angles in a triangle. It is written as: Sin(α)/a = Sin(β)/b = Sin(γ)/c. It is used to solve triangles when we know either two angles and one side, or two sides and one non-included angle.