9.2 Trigonometry: Tangent Ratio Day 1 Geometry 9.2 Trigonometry: Tangent Ratio Day 1

SOH-CAH-TOA In a right triangle, there are three sides, two acute angles, and a right angle From either acute angle, there is an opposite leg, an adjacent leg and the hypotenuse A leg of a right triangle is a side that forms the right angle The hypotenuse is always opposite the right angle

SOH-CAH-TOA A Hypotenuse Opposite Leg C Adjacent Leg B

SOH-CAH-TOA A Hypotenuse Adjacent Leg C Opposite Leg B

9.2 Tangent Ratio Tangent Ratio, Cotangent Ratio and Inverse Tangent Objectives Use the tangent ratio in a right triangle to solve for unknown sides lengths. Use the cotangent ratio in a right triangle to solve for unknown sides lengths. Relate the tangent ratio to the cotangent ratio. Use the inverse tangent in a right triangle to solve for unknown angle measures. SOH – CAH – TOA

9.2 Problem 2 Slope and Right Triangles Pg. 672 Collaborate 1-6 (8 Minutes) The TANGENT (TAN) of an acute angle in a right triangle is the ratio of the length of the side that is opposite the reference angle to the length of the side that is adjacent to the reference angle. Collaborate 7-10 (6 Minutes) Collaborate 11-13 (5 Minutes)

9.2 Problem 2 Slope and Right Triangles 12.

9.2 Problem 2 Slope and Right Triangles 13. Together #14

9.2 Problem 4 Cotangent Ratio The cotangent of an angle is the reciprocal of the tangent of an angle. The COTANGENT (COT) of an acute angle in a right triangle is the ratio of the length of the adjacent leg to the length of the opposite leg in relation to the angle used. We can always use the tangent function.

9.2 Problem 4 Cotangent Ratio On the calculator There is no cotangent button COT is the reciprocal of TAN cot 𝜃 = 1 tan 𝜃 Example 𝐹𝑖𝑛𝑑 cot 35 𝑜 = 1 tan 35 𝑜 ≈1.43

Review Tangent Day 1 What is the ratio of sides in a right triangle for the tangent function? Opposite over Adjacent What is the ratio of sides in a right triangle for the cotangent function? Adjacent over Opposite

9.2 Trigonometry: Tangent Ratio Day 2 Geometry 9.2 Trigonometry: Tangent Ratio Day 2

Review Tangent Day 1 What is the ratio of sides in a right triangle for the tangent function? Opposite over Adjacent What is the ratio of sides in a right triangle for the cotangent function? Adjacent over Opposite

9.2 Tangent Ratio Tangent Ratio, Cotangent Ratio and Inverse Tangent Objectives Use the tangent ratio in a right triangle to solve for unknown sides lengths. Use the cotangent ratio in a right triangle to solve for unknown sides lengths. Relate the tangent ratio to the cotangent ratio. Use the inverse tangent in a right triangle to solve for unknown angle measures. SOH – CAH – TOA

9.2 Problem 5 Inverse Tangent The inverse tangent (arc tangent or 𝑡𝑎𝑛 −1 ) is the measure of an acute angle whose tangent is x. The relationship of sides used is opposite to adjacent. The calculator has an inverse tangent function. Only used to find the missing angle.

9.2 Problem 5 Inverse Tangent Collaborate 1-3 (5 Minutes)

9.2 Problem 5 Inverse Tangent

9.2 Problem 5 Inverse Tangent 3.

Problem 6: Road Grades How can we write 8% a different way? 0.08 8 100 Collaborate 1-5 (5 Minutes)

The diagram is not “proportional” to 8% Problem 6: Road Grades 3. 𝑡𝑎𝑛 −1 0.06 = 𝑡𝑎𝑛 −1 6 100 ≈ 3.43 𝑜 4. tan 10 ≈0.176≈17.6% Should read 18% 5. No The diagram is not “proportional” to 8%

tan 𝜃 = 𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑥=2200 𝑓𝑡 Does it make sense that the two sides of the triangle are equal? Yes: Isosceles, Right Triangle 45 𝑜 Adjacent X Opposite

Use the given diagram to find the missing side 100 ft 28 𝑜 X tan 28 = 100 𝑥 𝑥 tan 28 =100 𝑥= 100 tan 28 ≈188.07 𝑓𝑒𝑒𝑡 Shortcut tan 28 = 100 𝑥 𝑆𝑤𝑖𝑡𝑐ℎ 𝑥= 100 tan 28 ≈188.07 𝑓𝑒𝑒𝑡

Formative Assessment SOH-CAH-TOA Skills Practice 9.2: The Tangent Ratio Vocabulary Pg. 677 (1-3) Problem Set Pg. 678-684 (1-43) odd tan 𝜃 = 𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝐿𝑒𝑔 𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝐿𝑒𝑔 SOH-CAH-TOA Check the MODE on the calculators The MODE must be in DEGREES