The value of cot(61°) is equivalent to the tan( _ ) ex) The value of sin(28°) is equivalent to the cos( _ ) In the triangle shown here, remember that angles α and β are complementary.įor any pair of complementary angles, a trig ratio and its co‐function counterpart have the same value. ALWAYS MAKE SURE YOU RATIONALIZE DENOMINATORS! (You’ll need to determine the missing value first.)Įx) Evaluate all 6 trigonometric ratios for the angle θ shown in the diagram. For sine and cosine’s reciprocals always remember to pair an ‘S’ with a ‘C’: Sine’s reciprocal is Cosecant Cosine’s reciprocal is Secant It should be easy to remember that tangent and cotangent are reciprocals.Įx) Evaluate all 6 trigonometric ratios for the angle θ shown in the diagram.
“ S O H ” stands for “ Sine = O / H ” “ C A H ” stands for “Cosine = A / H ” “ T O A ” stands for “Tangent = O / A ” The reciprocals need to be committed to memory. THE SIX TRIGONOMETRIC RATIOS ADJ* = adjacent leg to angle θ OPP* = opposite leg to angle θ HYP = hypotenuse (* The role of ADJ and OPP depend on which of the two acute angles is being used)Ī handy way to remember the main three ratios is to use SOH‐CAH‐TOA. NOT a function of the triangle! There are 6 such ratios you can create using the sides of a right triangle. These ratios are a function of the angle. The trigonometric functions are based on this premise: the ratio of any two sides of similar right triangles can be paired with the angle measures they share. The measure of each angle (using a protractor): Similar Right Triangles Two right triangles are similar if their matching sides are in the same proportion. Handy Pythagorean Triples: just keep in mind when using Pythagorean triples that the hypotenuse MUST be the longest side of the right triangle. TRAGIC mistake!) a = c 2 − b2 b = c 2 − a2 and all of its rearrangementsĬ = a2 + b2 (can’t simplify to c = a + b. A + B + C = 180° (True for all triangles) A and B are complementary angles ( A + B = 90° ) Pythagorean Theorem: a2 + b2 = c 2. Upper case A, B and C are its angles with C being the right angle. The sides a and b are referred to as the legs of the right triangle. In this lesson we primarily use the phrase trig ratios rather than trig functions, but this shift will happen throughout the unit especially as we look at the graphs of the trig functions in lessons 4.5 and beyond.4.3 & 4.8 Right Triangle Trigonometry Anatomy of Right Triangles The right triangle shown at the right uses lower case a, b and c for its sides with c being the hypotenuse. It is also important to emphasize that knowing for example that the sine of an angle is 7/18 does not necessarily imply that the opposite side is 7 and the hypotenuse is 18, simply that 7/18 represents the ratio of sides Some of the check your understanding questions are centered around this idea of interpreting decimals as comparisons (question 4 and 5). It is critical that students understand that even a decimal value can represent a comparison of two sides. The use of the word “ratio” is important throughout this entire unit. Give students time to wrestle through this idea and pose questions such as “How do you know sine will stay the same? Can you give me a convincing argument?” Formalize Later It is not immediately evident to them that they would not change by the same amount, thus altering the ratio.
Throughout this unit we will continue to point out that a decimal can also denote a comparison of two sides and not just one singular quantity.įor question 6, students are likely to say that the sine ratio will stay the same since both the opposite side and the hypotenuse are increasing. Fractions emphasize the comparison of sides and decimals emphasize the equivalence of the ratios. In question 4, make sure students write the answers as fractions and decimals. You may wish to project the lesson onto a screen so that students can see the colors of the sides if they are using black and white copies.
The goal of today’s lesson is that students grasp the concept that angles in a right triangle determine the ratio of sides and that these ratios have specific names, namely sine, cosine, and tangent. They consider the relative size of sides in a right triangle and relate this to the measure of the angle across from it. Students start unit 4 by recalling ideas from Geometry about right triangles.
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