From Pythagorean Identities to Law of Cosines: Key Trigonometric Formulas Every IB SL Math Student Should Know
In IB SL Math, students are expected to master these identities and formulas. In this blog post, we will discuss some of the essential trigonometric identities and formulas that you need to know for IB SL Math.
1. Pythagorean Identities
The Pythagorean identities are the most basic and essential identities in trigonometry. They are based on the Pythagorean theorem, which states that in a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
sin^2 θ + cos^2 θ = 1
1 + tan^2 θ = sec^2 θ
1 + cot^2 θ = csc^2 θ
These identities are useful in simplifying trigonometric expressions and solving trigonometric equations.
2. Angle Sum and Difference Identities
The angle sum and difference identities are used to find the trigonometric values of the sum or difference of two angles. They are as follows:
sin (α ± β) = sin α cos β ± cos α sin β
cos (α ± β) = cos α cos β ∓ sin α sin β
tan (α ± β) = (tan α ± tan β) / (1 ∓ tan α tan β)
These identities are useful in solving trigonometric equations, especially when the angles are not simple multiples of 90 degrees.
3. Double Angle Identities
The double angle identities are used to find the trigonometric values of double the angle. They are as follows:
sin 2θ = 2 sin θ cos θ
cos 2θ = cos^2 θ − sin^2 θ
tan 2θ = 2 tan θ / (1 − tan^2 θ)
These identities are useful in simplifying trigonometric expressions and solving trigonometric equations.
4. Half Angle Identities
The half-angle identities are used to find the trigonometric values of half the angle. They are as follows:
sin (θ/2) = ± √[(1 − cos θ) / 2]
cos (θ/2) = ± √[(1 + cos θ) / 2]
tan (θ/2) = ± √[(1 − cos θ) / (1 + cos θ)]
These identities are useful in solving trigonometric equations, especially when the angles are not simple multiples of 90 degrees.
5. Law of Sines and Law of Cosines
The Law of Sines and Law of Cosines are used to solve triangles, where the lengths of some sides and the measures of some angles are known. They are as follows:
Law of Sines: a / sin A = b / sin B = c / sin C
Law of Cosines: a^2 = b^2 + c^2 − 2bc cos A
b^2 = a^2 + c^2 − 2ac cos B
c^2 = a^2 + b^2 − 2ab cos C
These formulas are useful in solving practical problems involving triangles, such as finding the distance between two points or the height of a building.
In conclusion, trigonometric identities and formulas are essential tools in solving trigonometric problems. IB SL Math students need to master these identities and formulas to solve practical problems involving triangles. The Pythagorean identities, angle sum and difference identities, double angle identities, half-angle identities, and Law
