Lesson 5: Finding unknown angles using trigonometry

Year level

Stage 5

Duration

60 minutes

Focal Topic

Trigonometry A

Learning Goals

Lesson Aims

Finding unknown angles using trigonometry

Understanding the existence of the inverse operation for trigonometric ratios

Learning Intentions

  • Students use trigonometric ratios sine, cosine and tangent, to find the size of unknown angles in right-angled triangles including in degrees and minutes
  • Understand the concept of an inverse operation for trigonometric ratios

Success Criteria

  • Students are able to perform inverse operations $\sin^{-1}$, $\cos^{-1}$, $\tan^{-1}$ to make $\theta$ (unknown angle) the subject
  • Students are able to compute the value of $\theta$ in a calculator to degrees and minutes

Syllabus Links

MAO-WM-01
  • develops understanding and fluency in mathematics through exploring and connecting mathematical concepts, choosing and applying mathematical techniques to solve problems, and communicating their thinking and reasoning coherently and clearly
MA5-TRG-C-01
  • Applies trigonometric ratios to solve right-angled triangle problems

Content descriptor

  • Apply trigonometry to find the size of unknown angles in right-angled triangles including in degrees and minutes

ICT and LIT

ICT

None in this lesson

LIT

  • Using the “Inverse Trig Bingo” literacy game to practice inverse trig functions while using correct mathematical terms.

Prerequisite Knowledge

  • Students should be able to solve linear equations of up to 2 steps and quadratic equations of the form $ax^2 = c$ (MA4-EQU-C-01)
  • Students should be able to express the trigonometric ratios sine, cosine and tangent, in terms of sides Opposite, Adjacent, Hypotenuse, for a given angle

Resources

Whiteboard/Smartboard (markers), student worksheet 5, Bingo cards, calculators

Vocabulary List

  1. Inverse operation – an operation that reverses the effect of another
  2. Inverse trigonometric function – $\sin^{-1}$, $\cos^{-1}$, $\tan^{-1}$, used to find angles
  3. Make $\boldsymbol{\theta}$ the subject – rearranging the equation so that $\theta$ is isolated
  4. Calculate in degrees and minutes – find angles in sexagesimal system (° ′ ″)
  5. Principal value – the main solution given by a calculator for an angle

Lesson Structure

Introduction

5 minutes

  1. Greet and settle students down
  2. Give time for students to prepare for the lesson
    • This includes giving time to take out resources for the lesson
  3. Teacher displays the discussion question from the previous lesson to make connections between lessons
    • “We have learnt to find the unknown side for a given angle using trigonometric ratios. Do you believe then that we could find an unknown ANGLE when given values for the ratios?”
  4. Tell students the answer is ‘yes’ and give a brief outline of learning outcomes for the lesson.

Body 1

Concept teaching

15 minutes

  1. Finding an unknown angle
    • Explain that finding an unknown angle in a right-angled triangle takes the same first step as finding an unknown side: Establishing an equation for trigonometric ratios and substituting values where possible/given.
    • Remind students $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}, \, \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}, \, \tan \theta = \frac{\text{opposite}}{\text{adjacent}}$
  2. Teacher introduces inverse trigonometric operations by following the steps below
    1. When solving equations such as $x - 4 = 8$, in order to solve for $x$, we need $x$ to be the subject. Since there is a $-4$ on the side that $x$ is on, we remove $-4$ by adding $4$ to both sides of the equation, preserving equality. Hence $x = 12$
    2. When solving $2x = 4$, $2$ is multiplied to $x$, hence removing the $2$ will require the opposite of multiplication, which is division by $2$. Hence $x = 2$
    3. When solving $x^2 = 4$, we square root the $x^2$ in order to make $x$ the subject. Hence $x = 2$, or $-2$.
    4. When we want $\theta$ to be the subject in trigonometric ratios equations, we perform an INVERSE trigonometric operation on both sides. This can be done by computing in calculators.
      • $\sin \theta = \frac{O}{H} \implies \sin^{-1}(\sin \theta) = \sin^{-1}(\frac{O}{H})$
      • $\theta = \sin^{-1}(\frac{O}{H})$
  3. Literacy game: “Inverse Trig Bingo” (LIT)
    1. Teacher prepares Bingo cards with answers (angles in degrees/minutes).
    2. Teacher calls out the problems from Teacher call-out problems verbally (e.g., “Find $\theta$ if $\sin \theta = 0.6$”) using full sentences.
    3. Students listen carefully, compute the answer, and mark it on their card.
    4. When a student gets a line, they explain to the class using correct vocabulary (Opposite, Hypotenuse, $\theta$).

Body 2

Independent practice

35 minutes

  1. Students work independently on worksheet 5.

Students are expected to complete the exercise for homework if they have not finished in class

Teacher roams around the classroom to help individually students with questions.

  • If many require explanation on a specific question, teacher brings attention from whole class to solve together the question on the board

Conclusion

5 minutes

Tell the students to pack up and stand behind their chairs.

Bingo cards

Download EDMT5663 Assignment 2 Bingo cards (PDF)

Teacher call-out problems

Call out each of the following problems using full sentences, for example, “Find $\theta$ if $\sin \theta = 0.6$”.

  1. $\sin \theta = 0.5$
  2. $\cos \theta = 0.707$
  3. $\tan \theta = 1.333$
  4. $\sin \theta = 0.6$
  5. $\cos \theta = 0.642$

Student worksheet 5

Download EDMT5663 Assignment 2 Lesson 5 Worksheet (PDF)

Student worksheet 5 solutions