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Warm-up:

Warm-up:. A surveyor standing 45 ft. from the base of a tree measures the angle of elevation to the top of the tree as 65.8 °. How tall is the tree?

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Warm-up:

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  1. Warm-up: • A surveyor standing 45 ft. from the base of a tree measures the angle of elevation to the top of the tree as 65.8°. How tall is the tree? • An airplane flying at an altitude of 6 miles is on a path that passes directly over an observer. The angle of depression from the plane to the observer is 53°. Find the distance from the observer to the plane.

  2. Warm-up (continued): 3. The cable supporting a ski lift rises 5 feet for each 18 feet of horizontal length. The top of the cable is fastened 870 feet above the cable’s lowest point. Find the lengths b and c, and the measure of angle theta.

  3. Answers 100.130 Feet D= 7.513 miles b = 3132 c = 3250.588 θ = 15.524º

  4. PrecalculusLesson ( ) Check: p. 317 # 45-66 (x3), 76-88 even

  5. Objective: • Evaluate trig functions of any angle. Unit Circle Quiz Quad 1

  6. 4.4 Day 2 Find trig ratios of any angle Angles in Standard Position

  7. Finding the Trig Ratios of an Angle in Standard Position Choose a point (x, y) on the terminal side and calculate the primary trig ratios. P(x, y) r y q x r2 = x2 + y2 x2 = r2 - y2 y2 = r2 - x2

  8. Finding the Trig Ratios of an Angle in Standard Position P(x, y) r y q x Note that x is a negative number r2 = (x)2 + y2 Remember that in quadrant II, x is negative so cosine and tangent will be negative. (x)2 = r2 - y2 y2 = r2 - (x)2

  9. Finding the Trig Ratios of an Angle in Standard Position The point P(3, 4) is on the terminal side of q . List the trig ratios and find q . P(3, 4) 5 4 q 3 r2 = x2 + y2 = 32 + 42 = 9 + 16 = 25 r = 5 q = 530

  10. Example 1 (a) The point P(-3, 4) is on the terminal side of q . List the trig ratios and findq . P(-3, 4) 5 4 q qref= 530 -3 Reference Angle r2 = x2 + y2 = (-3)2 + (4)2 = 9 + 16 = 25 r = 5 Principal Angle 1800 - 530 =1270 q = 1270

  11. Example 1 (b) The point P(-2, 3) is on the terminal side of q . List the trig ratios and findq . P(-2, 3) 3 qref= 560 q Reference Angle !! from your calculator -2 r2 = x2 + y2 = (-2)2 + (3)2 = 4 + 9 = 13 r = √ 13 Principal Angle 1800 - 560 = 1240 q= 1240

  12. Example 2Function ValueConstraint • sin θ = cos θ < 0 • Sec θ = -4 cot θ > 0

  13. Classwork Without a calculator P. 317 # 93, 95 Calculator practice Make sure you are in the correct mode P. 317 # 75-89 odd

  14. Using the ASTC Rule (All Students Take Calculus) All Sine Find angle A, to the nearest degree: 00 ≤ A < 3600 1800 -q q 3600 -q 1800 +q Tangent Cosine Quadrants RA 340 1460 340 I II sin A = 0.5632 3070 cos A = 0.5986 530 I 530 IV 3030 1230 II IV 570 tan A = -1.5643 2210 cos A = -0.7542 410 II 1390 III 2400 IV III sin A = -0.8667 600 3000 310 2110 I III tan A = 0.5965 310

  15. Example 2Solve the equation.

  16. Assignments Classwork: Homework(4.4b ) P. 316 #12-16 even, 21-24 all, 38-44 even, 55, 65, 92, 96 Test 4.1-4.4 Friday

  17. Closure Use a calculator. Round to three decimal places. 1. cos 17º 2. sin 43º 3. cot 81º 4. sec 75º 5. Find the reference angle for 13°. 6. Find the reference angle for –98°.

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