Basic Trigonometry Sine Cosine Tangent

 The Language of Trig Opposite Hypotenuse target angle Adjacent The target angle is either one of the acute angles of the right triangle. Hypotenuse Opposite target angle  Adjacent

 Trig Equations Hypotenuse Opposite target angle  Adjacent opposite sin Hypotenuse

 Trig Equations Hypotenuse Opposite target angle  Adjacent Adjacent COS Hypotenuse

 Trig Equations Hypotenuse Opposite target angle  Adjacent opposite SOH - CAH - TOA

Given sides, find requested ratios: Locate indicated angle and classify sides in relation to it. Use proper equation and write the ratio. Use calculator to round to the nearest ten-thousandths place. Hypotenuse Opposite Adjacent Find sin, cos, and tan of A. 8.2 opposite sin A 24.5 Hypotenuse 23.1 COS A Adjacent Hypotenuse 24.5 8.2 opposite TAN A 23.1 Adjacent

Given sides, find requested ratios: Find sin A, cos A, tan A Find sin B, cos B, tan B. What do you notice? Hypotenuse Opposite Adjacent Opposite Adjacent 6.8 opposite 4 opposite sin A  0.9067 sin B  0.5333 7.5 Hypotenuse 7.5 Hypotenuse 4 COS A 6.8 Adjacent  0.5333 COS B Adjacent  0.9067 7.5 Hypotenuse Hypotenuse 7.5 6.8 opposite 4 TAN A  1.7000 opposite TAN B  0.5882 4 6.8 Adjacent Adjacent

Solving for the unknown: !!!!!! MAKE SURE YOUR CALCULATOR MODE IS SET TO DEGREES NOT RADIANS!!!!!!!!! Unknown on top…..Multiply x sin 40 24 Calculator: 24 * (sin 40)  15.4

Solving for the unknown: Unknown on bottom…..Divide 24 sin 40 x Calculator: 24 (sin 40)  37.3

Solving for the unknown: Unknown angle…..use inverse 15.4 x sin 24 Calculator: 2nd sin-1 (15.4/24)  39.9

h h = 52.6(sin 30)  26.3 Solving problems using trig: 52.6 Hypotenuse Choose your target angle and classify sides in relation to it. Use proper function and write the equation placing the unknown wherever it falls. Solve…Unless directed otherwise, round sides and angles to nearest tenth unless told otherwise. Hypotenuse Opposite sin Opposite Hypotenuse h sin 30  52.6 h = 52.6(sin 30)  26.3

x x = 6.1 (tan 41)  7.0 in Solving problems using trig: 6.1 Opposite Adjacent Opposite Tan Adjacent 6.1 Tan 41 x x = 6.1 (tan 41)  7.0 in

x x = 2nd COS (84/130)  50 Solving problems using trig: 84 130 Hypotenuse Adjacent Hypotenuse cos Adjacent x 84 cos 130 x = 2nd COS (84/130)  50

Angle of Elevation The angle of elevation is always Measured from the Horizontal UP… It is usually INSIDE the triangle.

Angle of Depression x The angle of depression is always Measured from the Horizontal DOWN… It is usually OUTSIDE the triangle. x   However….because horizontal lines are parallel, an angle of depression is equal to its alternate interior angle of elevation.

h = 30(tan 67) = 70.7m Practice h h opposite Opposite TAN 67 Adjacent

x  16 ft Adj Hyp Opp x ft x ft 32° 32° 25 ft 25 ft TAN From a point on the ground 25 feet from the foot of a tree, the angle of elevation of the top of the tree is 32º. Find to the nearest foot, the height of the tree. Adj Hyp Opp TAN x ft x ft 32° 32° x  16 ft 25 ft 25 ft

Example 2: From the top of a tower 60 feet high, the angle of depression to an object on the ground is 35. Find the distance from the object to the base of the tower to the nearest foot. 35° 60 ft 60 ft TAN 35 x 35° x  86 ft x

Example 3: In an isosceles triangle, the base is 28 cm long, and the legs are 17 cm long. Find the measure of a base angle to the nearest degree. 17 cm 28 cm x° 14 cos x 28/2=14 cm 17 x  35

Example 4: The height of a flagpole is 12 meters Example 4: The height of a flagpole is 12 meters. A student stands 50 meters from the foot of the flagpole. What is the measure of the angle of elevation from the ground to the top of the flagpole to the nearest degree? 12 m 12 m TAN x 50 m x° x  13 50m