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IBMathsDPApplications & Interpretation (AI)HLRevision Notes3. Geometry & TrigonometryTrigonometryPythagoras & Right-Angled Trigonometry

Pythagoras & Right-Angled Trigonometry (DP IB Applications & Interpretation (AI)): Revision Note

Amber

Written by: Amber

Reviewed by: Dan Finlay

Updated on

DP Applications & Interpretation (AI): HLDP Applications & Interpretation (AI): SLDP Analysis & Approaches (AA): HLDP Analysis & Approaches (AA): SL

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Pythagoras

What is the Pythagorean theorem?

  • Pythagoras’ theorem is formula using the lengths of the sides of a right-angled triangle

  • The square of the hypotenuse is equal to the sum of the squares of the two shorter sides

  • The formula can be written as a squared plus b squared equals c squared

    • a and b are the lengths of the shorter sides

    • c is the length of the hypotenuse

  • To find the length of the hypotenuse:

    • Use c equals square root of a squared plus b squared end root

  • To find the length of a shorter side:

    • Use a equals square root of c squared minus b squared end root

  • The converse of the theorem is true

    • If a squared plus b squared equals c squared

    • Then the triangle has a right-angle

Right-angled triangle with sides labelled a, b, and hypotenuse c; angle between a and b is a right angle.
Example of right-angled triangle

Examiner Tips and Tricks

The formula for Pythagoras' theorem is not given in the formula booklet.

Worked Example

ABCDEF is a chocolate bar in the shape of a triangular prism.  The end of the chocolate bar is an isosceles triangle where AC = 3 cm and AB = BC = 5 cm.  M is the midpoint of AC. This information is shown in the diagram below.

diagram-for-we-3-3-1-pythag

Calculate the length BM.

3-3-1-ai-sl-pythag-we-solution

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Right-Angled Trigonometry

What is right-angled trigonometry?

  • Right-angled trigonometry is used to find the angles and the lengths of the sides of a right-angled triangle

  • Three trigonometric functions are used to show the ratios between different pairs of sides

    • sine (sin)

    • cosine (cos)

    • tangent (tan)

What are the formulas for sin, cos and tan?

  • You need to label one angle and all three sides of the right-angled triangle

    • theta is one of the acute angles

    • H is the length of the hypotenuse

    • O is the length of the side opposite theta

    • A is the length of the side adjacent to theta

Two right-angled triangles; left triangle with angle θ, hypotenuse H, opposite O, adjacent A. Right triangle similar, rotated counterclockwise.
Example of labelling a right-angled triangle

  • The formulas for sin, cos and tan are:

    • sin space theta blank equals space opposite over hypotenuse space equals space O over H

    •  cos space theta blank equals space adjacent over hypotenuse space equals space A over H

    • tan space theta blank equals space opposite over adjacent space equals space O over A

Diagram of a right-angled triangle with labels for hypotenuse, opposite, and adjacent sides; includes trigonometric ratios: sine, cosine, tangent.

Examiner Tips and Tricks

The mnemonic SOHCAHTOA is often used as a way of remembering the formulas. These are not given in the formula booklet.

How can I use SOHCAHTOA to find missing lengths and angles?

  • STEP 1
    Label the sides of the triangle as H, O and A

  • STEP 2
    Identify which trigonometric ratio to use: sin, cos or tan

    • Write down the letters of the length you are given or want to find

    • Find the two letters in SOHCAHTOA to identify which ratio to use

      • If you have A and H then use cos

  • STEP 3
    Substitute the values into the relevant trigonometric formula

    • Remember to put brackets around the angle

      • sin open parentheses 50 close parentheses equals straight A over 7  or  cos open parentheses 40 close parentheses equals 3 over straight H or tan open parentheses theta close parentheses equals 3 over 7

  • STEP 4
    Rearrange and solve for the unknown letter

    • You will either need to multiply or divide to find a missing length

      • sin open parentheses 50 close parentheses equals straight A over 7 leads to straight A equals 7 cross times sin open parentheses 50 close parentheses

      • cos open parentheses 40 close parentheses equals 3 over straight H leads to straight H equals fraction numerator 3 over denominator cos open parentheses 40 close parentheses end fraction

    • You will need to use the inverse trigonometric functions to find a missing angle

      • tan open parentheses theta close parentheses equals 3 over 7 leads to theta equals tan to the power of negative 1 end exponent open parentheses 3 over 7 close parentheses

Worked Example

Find the values of x and y in the following diagram. Give your answers to 3 significant figures.

sa-diagram-for-we-3-3-1-trig
3-3-1-ai-sl-r-a-trig-we-solution

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3D Problems

Can I use Pythagoras' theorem and trigonometry in 3D?

  • You can break a 3D problem into 2D triangles

  • The formula for the length of the diagonal of a cuboid with dimensions a cross times b cross times c is d equals square root of a squared plus b squared plus c squared end root

    • This is sometimes referred to as the 3D version of Pythagoras' theorem

  • You can find missing angles by drawing right-angled triangles

    • You can find angles between lines and planes

Diagram showing the angle between a line PQ and planes in a 3D box, labelled with points, lines, and angles x°, y°, z°. Includes explanatory text.

Examiner Tips and Tricks

Draw a big enough sketch so that you can clearly annotate it with missing angles and lengths.

Worked Example

A pencil is being put into a cuboid shaped box. The base of the box has a width of 4 cm and a length of 6 cm. The height of the box is 3 cm. Find:

a) the length of the longest pencil that could fit inside the box,

ai-sl-3-3-1-3d-pythag-trig-we-solution-a

 

b) the angle that the pencil would make with the top of the box.

ai-sl-3-3-1-3d-pythag-trig-we-solution-b

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    Author
    Reviewer
    Amber

    Author: Amber

    Expertise: Maths Content Creator

    Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.

    Dan Finlay

    Reviewer: Dan Finlay

    Expertise: Maths Subject Lead

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.