VIEWS: 37 PAGES: 7 CATEGORY: Education POSTED ON: 12/24/2009 Public Domain
5.2 RIGHT TRIANGLE TRIGONOMETRY Trigonometry is study involving triangle measurement. Right triangles are used to study functions that involve trigonometry (trigonometric functions) Trigonometric function is a ratio of any two sides of a right angle triangle. Hypotenuse(r) Opposite(y) θ Adjacent(x) x2 + y2 = r 2 (Pythagoras theorem) The 6 ratios defined for a right triangle are: y x y r sin θ = , cos θ = , tan θ = , csc θ = , r r x y r x sec θ = cot θ = x y Example 1 5 Let θ be an acute angle such that sin θ = find: 3 a) secθ + tan θ b) ( cot θ ) + ( csc θ ) 2 2 Solution y ( ) 2 a) sin θ = ⇒ 32 = 5 + x 2 ⇒ x = 2 r r 3 y 5 ⇒ sec θ = = and ⇒ tan θ = = x 2 x 2 3+ 5 ⇒ sec θ + tan θ = 2 x 2 4 b) cot θ = = ⇒ ( cot θ ) = 2 y 5 5 2 r 3 9 csc θ = ⇒ ( csc θ ) = 2 = y 5 5 4 9 13 ⇒ ( cot θ ) + ( csc θ ) = 2 2 + = 5 5 5 Trigonometric ratios of special acute angles The equilateral and isosceles triangles shown below illustrates how trigonometric ratios for angles 30° 45° and 60° are derived. 30°30° x h x x x 60° 2 2 60° Note: h = x 2 − ( 2 )2 = x 3x 2 x 3x 1 h 3 sin 30° = 2 = , sin 60° = = 2 = x 2 x x 2 x 1 h 3 cos 60° = 2 = , cos 30° = = x 2 x 2 r y 45° y y y y 1 2 sin 45° = cos 45° = = = = = r y2 + y2 y 2 2 2 3 Using the above two triangles, the following are obtained; 1. sin 0° = cos 90° = 0 1 2. sin 30° = cos 60° = 2 1 2 3. sin 45° = cos 45° = = 2 2 3 4. sin 60° = cos 30° = 2 5. sin 90° = cos 0° = 1 6. tan 0° = cot 90° = 0 3 7. tan 30° = cot 60° = 3 8. tan 45° = cot 45° = 1 9. tan 60° = cot 30° = 3 Example 2 Find the exact values of; a) 3 sin 45° cos 30° − cot 30° π π π b) 2 tan + cos csc 4 3 6 π π π c) 2 cos cot + 2 3sec 3 4 6 Solution 2 3 3 6 −4 3 a) 3sin 45° cos 30° − cot 30° = 3. . − 3= 2 2 4 π π π 1 b) 2 tan + cos csc = 2.1 + .2 = 3 4 3 6 2 π π π 1 2 3 2 +8 c) 2 cos cot + 2 3sec = 2. .1 + 2 3. = 3 4 6 2 3 4 4 Application of Trigonometric Functions of Acute Angles Angle of Elevation is an angle measured upwards from the horizontal as shown in the figure below θ Angle of depression is an angle measured downwards from the horizontal as shown below θ 5 Example 3 A 5 meters ladder is resting against a wall and makes an angle of 30° with the ground. Find the height to which the ladder will reach the wall. Solution 5 h 30° h 1 = sin 30° = ⇒ h = 2.5m 5 2 Example 4 Find the height of a building if the angle of elevation from the top of the building changes from 60° to 30° as the observer moves 30m further from the building. Solution h 30° 60° 30 x h = tan 60° = 3 ⇒ h = x 3 x h 3 30 x 3 30 3 = tan 30° = ⇒h= + x + 30 3 3 3 x 3 30 3 ⇒x 3= + 3 3 x 3 30 3 ⎛ 3 ⎞ 30 3 ⇒ x 3− = ⇒ x⎜ 3 − ⎜ ⎟= 3 3 ⎝ 3 ⎟⎠ 3 6 ⎛ 3 ⎞ 30 3 ⇒ x⎜ 3 − ⎜ ⎟= ⎝ 3 ⎟⎠ 3 2 3 30 3 ⇒ x. = 3 3 ⇒ x = 15 3 ⇒ h = 15 3. 3 = 45 Example 5 Find the exact height h of the triangle below hm 30° 60° 5 3 m Solution hm 30° 60° 5 3 m x h = tan 60° = 3 ⇒ h = x 3 x h 3 x 3 = tan 30° = ⇒h= +5 x+5 3 3 3 7 x 3 ⇒x 3= +5 3 ⎛ 3⎞ ⇒ x⎜⎜ 3− ⎟=5 ⎝ 3 ⎟⎠ 2 3 ⇒ x. =5 3 5 3 ⇒x= 2 5 3 15 ⇒h= . 3 = = 7.5m 2 2