Circles and Right Triangle Trigonometry

II. Basic TrigonometryCIRCLES AND RIGHT TRIANGLE TRIGONOMETRY (Ch. 4) I. CIRCLES: circle- the set of all points that are equidistant from a point (the center). radius(r) distance from the center to the circle. diameter a chord that passes through the center (d=2r) Circumference (c) the distance around the edge of the circle diameter [\Pi=\frac{c}{d}\approx3.141592653589...] II. ARC LENGTH: • formula for finding arc length: [s=r\theta] 5- arc length r-radius -angle measure (rad) 2. \(d=4ft\) 3. \(s=15m\) \[S=7.5ft\] \[\theta=2.5~rad\] find r. 1. \(r=12in\) \[\theta=63^{\circ}\] \[63^{*}=\frac{7\pi}{20}rad\] given \(d=2r\) given \(s=r\theta\) \[r=2ft\] \[s=(12)(\frac{7\pi}{20})\] given \(s=r\theta\) \[s=\frac{21w}{5}in\] \[7.5=2\theta\] \[\theta=3.75rad.\] given \(s=r\theta\) \[15=2.5r\] \[r=6m\] III. TRIANGLES: •in any triangle, the sum of the interior angles is always \(180^{\circ}/\pi\) in a right triangle the acute angles are always complementary 4. Find x 5. find x \[\frac{\pi}{2}=\frac{2\pi}{7}-x\] \[X=\frac{3\pi}{14}\] \[180^{\circ}=54^{\circ}+68^{\circ}\cdot x^{\circ}\] \[x^{\prime}=58^{\circ}\] X 68 \[\frac{2\pi}{7}\] 54 X circumference IV. PYTHAGOREAN THEOREM: 6. a triangle has side lengths of 7. given that the hypotenuse 5, 11, and 12. classify the triangle of a right righ triangle measures by its angles. 17 and a leg measures 8, find the length of the other leg. \[12^{2}<5^{2}+11^{2}\] acute \[144<25+121\] \[144<146\] in a right triangle 8. find x \[34^{2}=21^{2}+x^{2}\] \[1156=441-x^{2}\] \[17^{2}=8^{2}+b^{2}\] \[289=b4+b^{2}\] \[225=b^{2}\] \[b=15\] \[715=\underline{x^{2}}\] 15 34 21 \[x=\sqrt{715}\] \[x\approx26.7\] X V. RIGHT TRIANGLE TRIGONOMETRY: \[sin~\theta=\frac{opposite}{nypotenuse}\] \[cos~\theta=\frac{od\mu acent}{hypotenuse}\] \[tan~\theta=\frac{cpposite}{ad,acen~t}\] \[csc~\theta=\frac{hupotenuse}{bpposite}\] \[sec~\theta=\frac{hupoo~tenuse}{ad_{1}acent}\] \[\theta=\frac{adiacent}{opposite}\] cot adjacent 9. use the diagram to find each trigonometric ratio a. \(sin~\theta\boxed{\frac{12}{13}}\) d. \(csc~\theta\) 13 12 13 5 b. \(cos\theta\) \[\frac{5}{13}\] e. \[\frac{13}{5}\] \[sec~\theta\] 12 c. tane \[\frac{12}{5}\] f. \[\frac{5}{12}\] \[cot~\theta\] use inverse trigonometric functions on ratios to find an angle sin cos tan, \(csc^{-1},\) sec, cot hypotenuse θ opposite Use right triangle trigonometry to solve the following for x: [cos x = \frac{3}{7}] [x = cos^{-1}(\frac{3}{7})] [x \approx 65^{\circ}] [sin x = \frac{10}{19}] [x = sin^{-1}(\frac{10}{19})] [x \approx 31^{\circ}] [tan 25 = \frac{15}{x}] [x = \frac{15}{tan 25}] [x \approx 32.2] VI. ANGLE OF ELEVATION/ DEPRESSION: A surveyor is standing 75 feet from the base of a large flagpole. The surveyor measures the angle of elevation to the top of the pole as 52. How tall is the flagpole? [tan 52 = \frac{x}{75}] [75 * tan 52 = x] [x \approx 96 ft]   A navigator aboard a passenger jet measures the angle of depression to a tiny island to be 12 degrees. The line of sight from the plane to the island is x miles. If the jet is flying at an altitude of 33,000 feet, then what is the horizontal distance to the island? Give the answer in miles. [tan 12 = \frac{33000}{x}] [x = \frac{33000}{tan 12}] [x \approx 155252.8 ft] [x \approx 29.4 mi]