Trigonometric Ratio
What is Trigonometric Ratio?
A triangle in which one angle is a right angle (90 degrees).
Key formula / rule: Sine
Key points
- Define and calculate the six trigonometric ratios for a given right-angled triangle.
- Recall and apply the values of trigonometric ratios for standard angles.
- Understand and use fundamental trigonometric identities.
- Solve problems involving heights and distances using trigonometric ratios.
Common exam trap
Confusing opposite and adjacent sides.
Definitions
- Term
Right-angled triangle
- Meaning
A triangle in which one angle is a right angle (90 degrees).
- Term
Hypotenuse
- Meaning
The side opposite the right angle in a right-angled triangle; it is the longest side.
- Term
Opposite side
- Meaning
The side opposite to the given angle in a right-angled triangle.
- Term
Adjacent side
- Meaning
The side next to the given angle, which is not the hypotenuse, in a right-angled triangle.
- Term
Trigonometric Ratios
- Meaning
Ratios of the lengths of the sides of a right-angled triangle with respect to its angles.
Learning objectives
Define and calculate the six trigonometric ratios for a given right-angled triangle.
Recall and apply the values of trigonometric ratios for standard angles.
Understand and use fundamental trigonometric identities.
Solve problems involving heights and distances using trigonometric ratios.
Simplify trigonometric expressions.
Formulae
- Name
Sine
- Note
Ratio of opposite side to hypotenuse.
- Expression
sin $\θ$ = Opposite / Hypotenuse
- Name
Cosine
- Note
Ratio of adjacent side to hypotenuse.
- Expression
cos $\θ$ = Adjacent / Hypotenuse
- Name
Tangent
- Note
Ratio of opposite side to adjacent side.
- Expression
tan $\θ$ = Opposite / Adjacent
- Name
Cotangent
- Note
Reciprocal of tangent.
- Expression
cot $\θ$ = Adjacent / Opposite
- Name
Secant
- Note
Reciprocal of cosine.
- Expression
sec $\θ$ = Hypotenuse / Adjacent
- Name
Cosecant
- Note
Reciprocal of sine.
- Expression
csc $\θ$ = Hypotenuse / Opposite
- Name
Pythagorean Identity
- Note
Fundamental trigonometric identity.
- Expression
sin² $\θ$ + cos² $\θ$ = 1
- Name
Tangent Identity
- Note
Expresses tangent in terms of sine and cosine.
- Expression
tan $\θ$ = sin $\θ$ / cos $\θ$
Prerequisites
Basic Geometry (Triangles, Pythagoras Theorem)
Basic Algebra
Understanding of Angles and Ratios.
Common mistakes
Confusing opposite and adjacent sides.
Incorrectly applying reciprocal identities.
Memorizing standard angle values incorrectly.
Errors in applying trigonometric identities.
Assuming ratios apply to non-right-angled triangles without proper context.
Keywords
Trigonometry
Right-angled triangle
Sine
Cosine
Tangent
Heights and Distances
Trigonometric Identities
Standard Angles
Practice preview
If sin A = 3/5, then what is the value of cos A?…
easy
If cos θ = 12/13, then what is the value of sin θ?…
easy
If tan θ = 3/4, then what is the value of (sin θ + cos θ)?…
medium