2.4 Applications of Trigonometry and Pythagoras Theorem

Note: if you cannot remember the trigonometric identities, revise notes for 2.3 Solving Triangles using Trigonometry.

Pythagoras’ Theorem

  • Pythagoras’ theorem governs the relationship between the lengths of the sides of a right-angled triangle:

a^{2}+b^{2}=c^{2}

Where c is the length of the hypotenuse and a and b are the lengths of the other sides (note that it doesn’t matter which side is chosen to be a and which is chosen to be b).

Picture 1

Example

Picture 2

We wish to find the length of the unknown side in the right-angled triangle above. As the unknown side is not the hypotenuse, we will first have to rearrange our formula to make either a or b the subject. In this case, we will choose a:

a^{2}+b^{2} =c^{2}

a^{2} =c^{2}-b^{2}

a =\sqrt{c^{2}-b^{2}}

Now, we substitute in the known values:

a=\sqrt{15^{2}-8^{2}}=12.69

Trigonometry in a Circle

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2.3 Solving Triangles using Trigonometry

Using Sine

  • The trigonometric identity for sine tells us that for a right-angled triangle:

\sin (\theta)=\frac{O}{H}

Where O is the length of the edge opposite the angle θ and H is the length of the hypotenuse.

Picture 2

Example

Picture 3

We wish to find the unknown angle, θ, in the above right-angled triangle. As we have the lengths of the hypotenuse and opposite side, we can use sine to do this:

\sin (\theta) =\frac{O}{H}=\frac{5}{7}

\theta =\sin ^{-1}\left(\frac{5}{7}\right)=45.58^{\circ}

Using Cosine

Read More »2.3 Solving Triangles using Trigonometry