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The word trigonometry means “triangle measurement.” You can use trigonometry to calculate the lengths of sides and the measures of angles in triangles. Trigonometry has been used for centuries in the study of: surveying astronomy engineering © The Visual Classroom geography physics B opposite A © The Visual Classroom adjacent C B adjacent A © The Visual Classroom opposite C B opposite A opp sin A hyp © The Visual Classroom adjacent adj cos A hyp C opp tan A adj B adjacent A opp sin B hyp © The Visual Classroom opposite adj cos B hyp C opp tan B adj B opp A adj C SOH CAH TOA © The Visual Classroom SOH CAH TOA B hy p 10 A 8 C 6 adj © The Visual Classroom op p B SOH CAH TOA hy p 5 adj A © The Visual Classroom 12 opp C Example 1: Determine side b SOH CAH TOA b tan B 8 b tan 55 8 b= b= © The Visual Classroom B 55º 8 cm A ad C j b opp Example 2: Determine side c if A is 31°. B hyp 13 cos 31 c A 13 cm C ad j SOH CAH TOA © The Visual Classroom Example 3: Determine side p if P is 45°. R Q SOH CAH TOA © The Visual Classroom P Ex. 4: From a distance of 20 m from the base a lighthouse the angle of elevation to the top of a lighthouse is 38º. Determine the height of the lighthouse. hh 38º 20 m The lighthouse is © The Visual Classroom m high. © The Visual Classroom B SOH CAH TOA opp A adj C opp sin A hyp adj cos A hyp opp tan A adj opp A sin hyp adj A cos hyp opp A tan adj 1 © The Visual Classroom 1 1 SOH CAH TOA 8 sin A 10 1 8 A sin 10 A 53 © The Visual Classroom 6 cos A 10 1 6 A cos 10 A 53 8 tan A 6 18 A tan 6 A 53 Example 1: Determine the measure of A. 8 tan A 13 tan A = 0.6154 B 8 cm opp A 13 cm A = tan-1(0.6154) ad j A = 31.6° © The Visual Classroom C SOH CAH TOA Example 2: Determine the measure of P. 12 cos P 17 R SOH CAH TOA cos P = 0.70588 P = cos–1(0.70588) Q P = 45.1 P © The Visual Classroom Example. 3: In DPQR, Q = 90°, PR = 8 cm and PQ = 4 cm. Find R. hy p 4 sin R 8 sin R 0.5 R sin R 30 © The Visual Classroom 1 R 0.5 P 4 cm Q op p SOH CAH TOA Example 4: Determine the measure of B B 3 cm 5 tan B 3 tan B = 1.6666 B = tan-1(1.6666) B = 59.0° © The Visual Classroom A 5 cm ad C j opp SOH CAH TOA Trigonometry Applications Problems 1. While walking to school you pass a barn with a silo. Looking up to the top of the silo you estimate the angle of elevation to the top of the silo to be about 14°. You continue walking and find that you were around 40 m from the silo. Using this information and your knowledge of trigonometric ratios calculate the height of the silo. © The Visual Classroom 2. A sailboat is approaching a cliff. The angle of elevation from the sailboat to the top of the cliff is 35°. The height of the cliff is known to be about 2000 m. How far is the sailboat away from the base of the cliff? © The Visual Classroom 3. A sailboat that is 2 km due west of a lighthouse sends a signal to the lighthouse that it is in distress. The lighthouse quickly signals a rescue plane that is 7 km due south of the lighthouse. What heading from due north should the plane take in order to intercept the troubled sailboat? © The Visual Classroom H/W: Follow HW log © The Visual Classroom