Trigonometry

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

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

sinθ=oppositehypotenuse=ac

The cosine of the given angle θ is given as

cosθ=adjacenthypotenuse=bc

360∘180∘90∘45∘30∘=2π radians=π radians=π2 radians=π4 radians=π6 radians

sin2θ+cos2θtan2θ+1cot2θ+1=1=sec2θ=cosec2θ

The cosine formula

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

cosAcosBcosC=b2+c2−a22bc=c2+a2−b22ca=a2+b2−c22ab

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

Area=s(s−a)(s−b)(s−c)−−−−−−−−−−−−−−−−−√

where,

s=a+b+c2

sin(a±b)cos(a±b)tan(a±b)=sinacosb±cosasinb=cosacosb∓sinasinb=tana±tanb1∓tanatanb

sin2Acos2Atan2A=2sinAcosA=cos2A−sin2A=2tanA1−tan2A

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