Before starting, you will need to understand how to use SOH CAH TOA. How to work out trig functions of angles without a scientific calculator Task 1 sine and cosine Work out sin 23 and cos 23 by constructing a right-angled triangle with a hypotenuse of length 100mm. (You will need to divide by the hypotenuse so 100mm is a good choice.) Draw the base: You will need a line approximately 150mm long. Draw the hypotenuse: Use a protractor to mark an angle of 23 from the left hand side of the base. Draw a line from that left hand end, through the 23 mark and make it 100mm long. (Be very accurate here.) www.teachitmaths.co.uk 2015 24081 Page 1 of 5
Draw the vertical (opposite) side: Use compasses to draw an arc from right-hand side of hypotenuse that crosses the base in two places. Then keeping the compasses the same size, draw another two arcs (one from each of the first two arcs). Draw a line from the end of the hypotenuse through the intersection of the arcs. To work out the value of sin 23, measure the length of the opposite side (it should be 39mm) and divide it by the length of the hypotenuse. This gives: sin 23 = 39 100 = 0.39 (2 d.p.) To work out cos 23, measure the length of the adjacent side (it should be 92mm) and divide it by the hypotenuse. This gives cos 23 = 92 100 = 0.92 (2 d.p.) Use the same method to work out sin 30 and cos 30 Check your answers with a scientific calculator. www.teachitmaths.co.uk 2015 24081 Page 2 of 5
Task 2 tangent Work out tan 18 by constructing a right-angled triangle with a 100mm base. Draw the base with length exactly 100mm. From the left-hand end of the base, measure an 18 angle and from the right-hand end of the base, measure a 90 angle. Complete the triangle by drawing a line from the left-hand end, through the 28 mark and from the right-hand end, through the 90 mark. To work out the value of tan 18, measure the length size of the opposite side (it should be 32mm) and divide by the length of the adjacent side (100mm). This gives tan 18 = 32 100 = 0.32 (2 d.p.) Use the same method to find tan 45 Check your answers with a scientific calculator. By using these two methods, it is possible to find the approximate trig value of any acute angle. www.teachitmaths.co.uk 2015 24081 Page 3 of 5
How to work out inverse trig functions without a scientific calculator Task 3 inverse sine Work out sin -1 0.68 First, draw a horizontal base. Next, draw a vertical line up from the right-hand end of the base. This side should be 68mm long (the value after decimal point, rounded to 2 d.p.). Then, at the top end of this vertical line pin the compass and draw an arc with a radius of 100mm. The point where the 100mm arc intersected the base line (if necessary, the base line could be extended) gives the left end of the horizontal base and the vertex of the unknown angle. Use the same method to find: sin -1 0.89 Task 4 inverse cosine Work out cos -1 0.68 There a no hints for this, apart from the fact that you will need to use a similar method to the one above. Use this method to work out cos -1 0.89 Task 5 inverse tangent Work out tan -1 1.42 First, draw the horizontal base line, exactly 100mm long. Then draw a vertical line up from the right end of the horizontal base line measuring 142mm. Next, join the top end of the vertical line with the left end of the horizontal base line. Finally, measure the size of the angle where the horizontal adjacent side joins the hypotenuse. The angle will be approximately 55 (rounded to nearest ). It is the angle with a fixed (this time 1.42) ratio of the opposite side and the adjacent side. By using this method, you can find any acute angle. www.teachitmaths.co.uk 2015 24081 Page 4 of 5
Teacher notes This is an excellent activity for deepening knowledge and understanding of trigonometry. It is designed as a student-led activity and should require minimal teacher input. An added bonus is the opportunity to apply construction skills. To increase the challenge, step-by-step diagrams have been included for the first investigation but not included in the subsequent investigations. Students could be asked to add these diagrams to the other investigations. www.teachitmaths.co.uk 2015 24081 Page 5 of 5