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Pythagoras' theorem

In a right-angled triangle, the area of the square on the hypotenuse is equal to sum of the areas of the squares on the two shorter sides.

In the above triangle

Square on side a + Square on side b = Square on side c

a^2+b^2=c^2

Trigonometric ratios in a right-angled triangle

With reference to the following following diagram:

The tangent of the given angle θ is given as

tanθ = oppositeadjacent=ab

The sine of the given angle θ is given as

²õ¾±²Ôθ=´Ç±è±è´Ç²õ¾±³Ù±ð³ó²â±è´Ç³Ù±ð²Ô³Ü²õ±ð=²¹³¦

The cosine of the given angle θ is given as

³¦´Ç²õθ=²¹»åÂá²¹³¦±ð²Ô³Ù³ó²â±è´Ç³Ù±ð²Ô³Ü²õ±ð=²ú³¦

Some common angles and their trigonometric ratios

360∘180∘90∘45∘30∘=2Ï€°ù²¹»å¾±²¹²Ô²õ=Ï€°ù²¹»å¾±²¹²Ô²õ=Ï€2°ù²¹»å¾±²¹²Ô²õ=Ï€4°ù²¹»å¾±²¹²Ô²õ=Ï€6°ù²¹»å¾±²¹²Ô²õ

Some trigonometric identities

²õ¾±²Ô2θ+³¦´Ç²õ2θ³Ù²¹²Ô2θ+1³¦´Ç³Ù2θ+1=1=²õ±ð³¦2θ=³¦´Ç²õ±ð³¦2θ

Triangle formulae

The cosine formula

When given three sides of a triangle, as shown below, we can find the angles using the cosine formula

³¦´Ç²õ´¡³¦´Ç²õµþ³¦´Ç²õ°ä=²ú2+³¦2−a22²ú³¦=³¦2+²¹2−b22³¦²¹=²¹2+²ú2−c22²¹²ú

The sine formula

We can use the sine formula to find a side given two sides and an angle which is not included between the given sides:

asinA=bsinB=csinC

The area of a triangle

In the triangle above, if two sides and an included angle are known, the area of the triangle can be found using the appropriate formula below:

Area=12absinCArea=12bcsinAArea=12casinB

If the angles are unknown and the three sides are known, the area can be determined using the formula

´¡°ù±ð²¹=²õ(²õ−a)(²õ−b)(²õ−c)−−−−−−−−−−−−−−−−−√

where,

s=a+b+c2

The addition formulae

²õ¾±²Ô(²¹Â±²ú)³¦´Ç²õ(²¹Â±²ú)³Ù²¹²Ô(²¹Â±²ú)=²õ¾±²Ô²¹³¦´Ç²õ²ú±³¦´Ç²õ²¹²õ¾±²Ô²ú=³¦´Ç²õ²¹³¦´Ç²õ²ú∓s¾±²Ô²¹²õ¾±²Ô²ú=³Ù²¹²Ô²¹Â±³Ù²¹²Ô²ú1∓t²¹²Ô²¹³Ù²¹²Ô²ú

Double angle formulae

²õ¾±²Ô2´¡³¦´Ç²õ2´¡³Ù²¹²Ô2´¡=2²õ¾±²Ô´¡³¦´Ç²õ´¡=³¦´Ç²õ2´¡âˆ’s¾±²Ô2´¡=2³Ù²¹²Ô´¡1−t²¹²Ô2´¡

Additional Resources

• - a set of pages from mathtutor.ac.uk covering all of the above plus exercises.
• - a Khan Academy video introducing the the trig ratios.
• - another explanation of the trig ratios from mathbff.
• - another video from mathbff explaining how to solve right triangles.