Angles of Elevation and Depression Trigonometry Angles of Elevation and Depression

Trigonometric Ratios Sine Opposite Hypotenuse Cosine Adjacent Tangent

Examples Find sin J, cos J, tan J, sin K, cos K, and tan K. Express each ratio as a fraction and as a decimal to the nearest hundredth.

Examples Find sin J, cos J, tan J, sin K, cos K, and tan K. Express each ratio as a fraction and as a decimal to the nearest hundredth. sin J = 5/13 = .38 cos J = 12/13 = .92 tan J = 5/12 = .42 sin K = 12/13 = .92 cos K = 5/13 = .38 tan K = 12/5 = 2.4

Examples Find x to the nearest hundredth.

Examples Find x to the nearest hundredth. tan 25 = x/18 18*tan 25 = x x = 8.39

Inverse Trigonometric Ratios Inverse trigonometric ratios give the measure of the angle. sin-1 x = m∠A cos-1 x = m∠A tan-1 x = m∠A

Examples Use a calculator to find the measure of ∠A to the nearest tenth.

Examples Use a calculator to find the measure of ∠A to the nearest tenth. cos A = 3/15 cos-1 (3/15) = A A = 78.5

Solving a Right Triangle To solve a right triangle, you need to know: Two side lengths or One side length and the measure of one acute angle

Examples Solve the right triangle. Round side measures to the nearest tenth and angle measures to the nearest degree.

Examples Solve the right triangle. Round side measures to the nearest tenth and angle measures to the nearest degree. HF = 12 F = sin-1 5/13 = 22.6 G = cos-1 5/13 = 67.4

Angle of Elevation An angle of elevation is the angle formed by a horizontal line and an observer’s line of sight to an object above the horizontal line.

Angle of Depression An angle of depression is the angle formed by a horizontal line and an observer’s line of sight to an object below the horizontal line.

Two angles of elevation or depression Angles of elevation or depression to two different objects can be used to estimate the distance between those objects. Similarly, the angles from two different positions of observation to the same object can be used to estimate the object’s height.

Examples Two buildings are sighted from atop a 200-meter skyscraper. Building A is sided at a 35° angle of depression, while Building B is sighted at a 36° angle of depression. How far apart are the two buildings to the nearest meter?

Examples Two buildings are sighted from atop a 200-meter skyscraper. Building A is sided at a 35° angle of depression, while Building B is sighted at a 36° angle of depression. How far apart are the two buildings to the nearest meter? Tan (35°) = x/200 X = 200 Tan (55°) X = 285.63 y – x = 285.63 – 275.28 Tan (36°) = y/200 y – x = 10 m Y = 200 Tan (54°) Y = 275.28

Examples How high is the disco ball?

Examples Set variable for unknown quantity: y Use variable to make a system of equations: tan 50 = x/y tan 40 = x/(5+y) Solve the system.

Examples y*tan 50 = x 5*tan 40 + y*tan 40 = x 5*tan 40 + y*tan 40 = y*tan 50 5*tan 40 = y*tan 50 – y*tan 40 5*tan 40 = y*(tan 50 – tan 40) y = (5*tan 40)/(tan 50 – tan 40) y = 11.9 x = 14.2

Examples To estimate the height of a tree she wants removed, Mrs. Long sights the tree’s top at a 70° angle of elevation. She then steps back 10 meters and sights the top at a 26° angle. If Mrs. Long’s line of sight is 1.7 meters above the ground, how tall is the tree to the nearest meter?

Examples Set variable for unknown quantity: y Use variable to make a system of equations: tan 70 = x/y tan 26 = x/(y + 10) Solve the system.

Examples Y tan 70 = x Y tan 26 + 10 tan 26 = x Y tan 70 = y tan 26 + 10 tan 26 Y tan 70 – y tan 26 = 10 tan 26 Y(tan 70 – tan 26) = 10 tan 26 Y = 10 tan 26 / (tan 70 – tan 26) Y = 2.16 X = 2.16 tan 70 X = 5.9 Height of tree = 5.9 + 1.7 = 7.6 m