Trigonometry Formula and Ratios 

Trigonometry Formula and Ratios

The “Toa Cah Soh” rule 



The “Toa Cah Soh” rule is used in trigonometry to calculate an unknown angle and length given either one of the following conditions:

a) Length of 2 sides of the triangle is given 

b) Or, 1 angle and 1 length of the triangle is given

In these conditions,
Tan Ø is used when there is opposite/adjacent  (O and A) 

Cos Ø is used when there is adjacent/hypotenuse (A and H)

Sin Ø is used when there is opposite/hypotenuse (O and H)

Sine Rule

Sine rule

Sine rule

Sine Rule 

a /Sin ㄥA = b / sin ㄥB = c / sin ㄥC 

Or Sin ㄥA / a = Sin ㄥB / b = sin ㄥ C/ c


Area of triangle (non-right angled triangle) using sine rule 

Area = ½ ab sin C = ½ ac sin B = ½ BC sin A

Cosine Rule 1

(With 2 sides of a triangle and the included angle)  

a2 = b2 + c2 – 2bc cos ㄥA

b2 = a2 +c2 – 2ab cos ㄥB

c2 = a2 +b2 – 2ab cos ㄥC

This will allow you to find the remaining side of the triangle. 

Cosine Rule 2

(With 3 sides of a triangle known to find an angle)  

Cos A = b2 + c2 – a2 / 2bc  

This will allow you to find an angle with all 3 sides known. 

Key Things to remember 

After memorising and remembering the above key factors of the trigonometry formula, sine rule, and cosine rule master the basics before heading to more complicated uses of trigonometry such as in the cosine differentiation.  

Memorize Basic Trigonometric Values: Know the values of sine, cosine, and tangent for common angles (0°, 30°, 45°, 60°, 90°) by heart. These values are frequently used in trigonometric calculations.

Understand the Unit Circle: The unit circle is a valuable tool for visualizing trigonometric functions. Familiarize yourself with the concept of the unit circle and how it relates to sine and cosine.

Trigonometric Identities: Learn and apply fundamental trigonometric identities such as the Pythagorean identities, co-function identities, sum and difference identities, and double angle identities.

Solving Trigonometric Equations: Practice solving trigonometric equations. Identify key strategies, such as factoring, using trigonometric identities, and applying inverse trigonometric functions.

Graphs of Trigonometric Functions: Understand the graphs of sine, cosine, and tangent functions. Learn to recognize their properties, including amplitude, period, and phase shifts.

Use Technology Wisely: Graphing calculators and computer software can be helpful, but avoid overreliance on them. Make sure you can solve trigonometric problems manually as well.Trigonometric Equations: Understand how to solve trigonometric equations and inequalities, particularly quadratic equations involving trigonometric functions.

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