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Defining Trigonometric Ratios

Define sine, cosine, tangent, and their reciprocals relative to an acute angle in a right triangle.

7 micro-lessons

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Naming the Sides

Introduce terms: Hypotenuse, Side Opposite, Side Adjacent.

Setting up the Right Triangle

When working with right triangles, correctly identifying the sides is the crucial first step to learning trigonometry. The hypotenuse is always the longest side, and it is positioned directly opposite the $90^\circ$ angle. Its position never changes, regardless of which acute angle you focus on.

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Triangle ABC with labeled sides

Right-angled triangle showing opposite and adjacent sides for Angle A.

A clean scientific diagram of a right-angled triangle ABC. The 90-degree angle is marked at vertex B. Vertex A is an acu…

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The Primary Ratios (sin, cos, tan)

Formulas for sine, cosine, and tangent of Angle A.

sin A = \frac{Opposite}{Hypotenuse}

= \frac{BC}{A C}

cos A = \frac{Adjacent}{Hypotenuse}

= \frac{A B}{A C}

tan A = \frac{Opposite}{Adjacent}

= \frac{BC}{A B}

Important: $sin A$ is a single mathematical symbol meaning 'the sine of angle A'. It is not the word 'sin' multiplied by 'A'. The same applies to all trigonometric ratios.

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The Reciprocal Ratios (cosec, sec, cot)

Formulas for cosecant, secant, and cotangent of Angle A.

Reciprocal of sin A

cosec A = \frac{Hypotenuse}{Opposite}

= \frac{1}{s i n A}

Reciprocal of cos A

sec A = \frac{Hypotenuse}{Adjacent}

= \frac{1}{c o s A}

Reciprocal of tan A

cot A = \frac{Adjacent}{Opposite}

= \frac{1}{t a n A}

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Does Triangle Size Matter?

Explain that trig ratios depend only on the angle, not the size of the triangle.

Similar Triangles Tell the Story

Imagine taking a point $P$ on the extended hypotenuse of right triangle $ABC$ and drawing a new line $PM$ perpendicular to the extended base $AB$ . This creates a new, larger right triangle, $PAM$ .

Because both triangles share angle $A$ and both have a $90^\circ$ angle, they are similar by the AA (Angle-Angle) similarity criterion ( $\Delta PAM \sim \Delta CAB$ ).

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Match Ratio to Definition

Match sin, cos, tan, etc., to their side-based definitions.

NoteMatch each trigonometric ratio to its correct side length definition.

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Terms

Definitions

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Checking Understanding: True or False

Determine if statements about abbreviations and definitions are correct.

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Is the following statement true or false? 'cos A is the abbreviation used for the cosecant of angle A.'

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Naming the Sides

Introduce terms: Hypotenuse, Side Opposite, Side Adjacent.

Setting up the Right Triangle

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Does Triangle Size Matter?

Explain that trig ratios depend only on the angle, not the size of the triangle.

Similar Triangles Tell the Story

Imagine taking a point $P$ on the extended hypotenuse of right triangle $ABC$ and drawing a new line $PM$ perpendicular to the extended base $AB$ . This creates a new, larger right triangle, $PAM$ .

Because both triangles share angle $A$ and both have a $90^\circ$ angle, they are similar by the AA (Angle-Angle) similarity criterion ( $\Delta PAM \sim \Delta CAB$ ).

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