Warm Up: No Calculators today! 2 6-25.  Copy the triangles at right onto your paper and label the missing side lengths and angle measures.  Then use the diagrams to write each trig ratio below as an exact value (fraction). 2 3 1 a.     sin 30º b.     cos 30º c.     tan 30º d.     sin 60º e.     cos 60º f.      tan 60º g.     sin 45º h.     cos 45º i.      tan 45º = 1 2 = 3 2 = 1 3 = 3 2 = 1 2 = 3 1 = 3 = 1 2 = 1 2 = 1 1 =1 Students come up and write answers Warm Up: No Calculators today!

6.1.3 Special Right Triangles and Trigonometry December 5, 2018 HW: 6-30 through 6-33

LO: SWBAT explain their thinking in proving trigonometric identities. Objectives CO: SWBAT use trigonometric ratios and their knowledge of special right triangles to prove a trigonometric identity. LO: SWBAT explain their thinking in proving trigonometric identities.

6-26. Examine the diagram at right. Write equations for sin(θ) and cos(θ). And write a relationship you know for the sides of every right triangle. sin 𝜃 = 𝑦 1 =𝑦 cos 𝜃 = 𝑥 1 =𝑥 𝑦 2 + 𝑥 2 = 1 2 Using your results from part (a) and your knowledge of right triangles, what is the value of (sin θ)2 + (cos θ)2 ? Explain how you know. (sin 𝜃 ) 2 +( cos 𝜃) 2 =1 Does this relationship appear to be true for all angles?  Use your values from problem 6-25 to calculate the sums below.  Write down your findings. (sin 30º)2 + (cos 30º)2 1 2 2 + 3 2 2 1 4 + 3 4 (sin 45º)2 + (cos 45º)2 1 2 2 + 1 2 2 1 2 + 1 2 Team

6-27. Testing examples as you did in part (c) of problem 6-26 can give you an idea of whether a statement is usually true.  But to test whether it is always true, you need to prove a general case.  Sometimes it helps to start with a theorem that you have already proved. Write an equation about the side lengths of ∆ABC at right and explain how you know it is true. 𝑎 2 + 𝑏 2 = 𝑐 2 because of the Pythagorean Theorem Now you will prove that (sin θ)2 + (cos θ)2 = 1. First write an equivalent expression for (sin θ)2 + (cos θ)2 using ∆ABC. 𝑎 𝑐 2 + 𝑏 𝑐 2 Now rewrite your expression from part (b) without parentheses.  Can you write it as a single fraction? 𝑎 2 + 𝑏 2 𝑐 2 What is your expression equal to?  Justify your answer. 1 if you divide both sides by 𝑐 2 in the Pythagorean Theorem. Team