Introduction to Trigonometric Formulas
Trigonometry, the branch of mathematics dealing with the relationships between the angles and sides of triangles, relies heavily on a set of core formulas. These formulas facilitate the calculation of unknown angles or sides, simplify expressions, and prove other mathematical identities. Mastery of these formulas is crucial for solving real-world problems in physics, engineering, astronomy, and computer science.
Basic Trigonometric Ratios
The foundation of trigonometry is the relationship between the angles and sides in a right-angled triangle.
Definitions of Ratios
In a right triangle with an angle θ:
- Sine (sin): The ratio of the length of the opposite side to the hypotenuse.
- Cosine (cos): The ratio of the length of the adjacent side to the hypotenuse.
- Tangent (tan): The ratio of the length of the opposite side to the adjacent side.
- Cosecant (csc): The reciprocal of sine.
- Secant (sec): The reciprocal of cosine.
- Cotangent (cot): The reciprocal of tangent.
Formulas for Basic Ratios
\[
\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}, \quad
\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}, \quad
\tan \theta = \frac{\text{opposite}}{\text{adjacent}}
\]
\[
\csc \theta = \frac{1}{\sin \theta} = \frac{\text{hypotenuse}}{\text{opposite}}, \quad
\sec \theta = \frac{1}{\cos \theta} = \frac{\text{hypotenuse}}{\text{adjacent}}, \quad
\cot \theta = \frac{1}{\tan \theta} = \frac{\text{adjacent}}{\text{opposite}}
\]
Reciprocal and Pythagorean Identities
These identities are fundamental in simplifying trigonometric expressions and solving equations.
Reciprocal Identities
- \(\csc \theta = \frac{1}{\sin \theta}\)
- \(\sec \theta = \frac{1}{\cos \theta}\)
- \(\cot \theta = \frac{1}{\tan \theta}\)
Pythagorean Identities
These identities relate the squares of sine and cosine:
- \(\sin^2 \theta + \cos^2 \theta = 1\)
- \(1 + \tan^2 \theta = \sec^2 \theta\)
- \(1 + \cot^2 \theta = \csc^2 \theta\)
Key Trigonometric Identities
These identities are essential tools for transforming and simplifying trigonometric expressions.
Angle Sum and Difference Formulas
These formulas express the sine, cosine, and tangent of sum or difference of two angles.
- Sine:
\[
\sin (A \pm B) = \sin A \cos B \pm \cos A \sin B
\]
- Cosine:
\[
\cos (A \pm B) = \cos A \cos B \mp \sin A \sin B
\]
- Tangent:
\[
\tan (A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}
\]
Double Angle Formulas
These relate the trigonometric functions of twice an angle to the functions of the original angle.
- Sine:
\[
\sin 2A = 2 \sin A \cos A
\]
- Cosine:
\[
\cos 2A = \cos^2 A - \sin^2 A = 2 \cos^2 A - 1 = 1 - 2 \sin^2 A
\]
- Tangent:
\[
\tan 2A = \frac{2 \tan A}{1 - \tan^2 A}
\]
Half-Angle Formulas
Useful for finding the sine, cosine, or tangent of half an angle.
- Sine:
\[
\sin \frac{A}{2} = \pm \sqrt{\frac{1 - \cos A}{2}}
\]
- Cosine:
\[
\cos \frac{A}{2} = \pm \sqrt{\frac{1 + \cos A}{2}}
\]
- Tangent:
\[
\tan \frac{A}{2} = \pm \sqrt{\frac{1 - \cos A}{1 + \cos A}} = \frac{\sin A}{1 + \cos A} = \frac{1 - \cos A}{\sin A}
\]
Product-to-Sum and Sum-to-Product Formulas
These identities are valuable for integrating and transforming trigonometric expressions.
Product-to-Sum Formulas
Convert products of sines and cosines into sums:
\(\sin A \sin B = \frac{1}{2} [\cos (A - B) - \cos (A + B)]\)
\(\cos A \cos B = \frac{1}{2} [\cos (A - B) + \cos (A + B)]\)
\(\sin A \cos B = \frac{1}{2} [\sin (A + B) + \sin (A - B)]\)
Sum-to-Product Formulas
Express sums of sines or cosines as products:
\(\sin A + \sin B = 2 \sin \frac{A + B}{2} \cos \frac{A - B}{2}\)
\(\sin A - \sin B = 2 \cos \frac{A + B}{2} \sin \frac{A - B}{2}\)
\(\cos A + \cos B = 2 \cos \frac{A + B}{2} \cos \frac{A - B}{2}\)
\(\cos A - \cos B = -2 \sin \frac{A + B}{2} \sin \frac{A - B}{2}\)
Special Trigonometric Values
Certain angles have well-known sine and cosine values, often used for quick calculations and proofs.
Common Angles and Their Values
- \(\theta = 0^\circ\) or \(0\) radians:
\[
\sin 0 = 0, \quad \cos 0 = 1
\]
- \(\theta = 30^\circ\) or \(\pi/6\):
\[
\sin 30^\circ = \frac{1}{2}, \quad \cos 30^\circ = \frac{\sqrt{3}}{2}
\]
- \(\theta = 45^\circ\) or \(\pi/4\):
\[
\sin 45^\circ = \frac{\sqrt{2}}{2}, \quad \cos 45^\circ = \frac{\sqrt{2}}{2}
\]
- \(\theta = 60^\circ\
Frequently Asked Questions
What are the basic trigonometric formulas I should memorize?
The basic trigonometric formulas include sine, cosine, tangent, cotangent, secant, and cosecant functions, along with their fundamental identities like sin²θ + cos²θ = 1, and tanθ = sinθ / cosθ.
What is the Pythagorean identity in trigonometry?
The Pythagorean identity is sin²θ + cos²θ = 1, which relates the sine and cosine of an angle.
What are the angle sum and difference formulas?
The angle sum and difference formulas are: sin(A ± B) = sinA cosB ± cosA sinB, and cos(A ± B) = cosA cosB ∓ sinA sinB.
How do I recall the double angle formulas?
Double angle formulas include: sin(2θ) = 2 sinθ cosθ, cos(2θ) = cos²θ - sin²θ, and tan(2θ) = 2 tanθ / (1 - tan²θ).
What are the half-angle formulas used for?
Half-angle formulas are: sin(θ/2) = ±√[(1 - cosθ)/2], cos(θ/2) = ±√[(1 + cosθ)/2], and tan(θ/2) = ±√[(1 - cosθ)/(1 + cosθ)].
What is the sum-to-product formula in trigonometry?
Sum-to-product formulas convert sums or differences of sines and cosines into products: sinA + sinB = 2 sin[(A+B)/2] cos[(A−B)/2], sinA - sinB = 2 cos[(A+B)/2] sin[(A−B)/2], etc.
How do I convert between degrees and radians in trig formulas?
To convert degrees to radians, multiply by π/180; to convert radians to degrees, multiply by 180/π.
What is the reciprocal identities in trigonometry?
Reciprocal identities are: cscθ = 1/sinθ, secθ = 1/cosθ, and cotθ = 1/tanθ.
Are there any useful reduction formulas?
Yes, reduction formulas simplify trigonometric functions of multiple angles, such as sin(π/2 - θ) = cosθ, and cos(π/2 - θ) = sinθ.
Where can I find a comprehensive list of trigonometric formulas?
A comprehensive list can be found in trigonometry textbooks, online math resources, and dedicated formula sheets for quick reference during studies or exams.
