Solutions Of Right Triangles PDF

MECHNOLOGYC ANDALUYONG S OLLEGE OF T CIENCE & SOLUTION OF RIGHT TRIANGLE Precalculus PREC111 MECHNOLOGYC ANDALUYONG OLLEGE OF S T CIENCE & O...

MECHNOLOGYC ANDALUYONG S OLLEGE OF T CIENCE & SOLUTION OF RIGHT TRIANGLE Precalculus PREC111 MECHNOLOGYC ANDALUYONG OLLEGE OF S T CIENCE & OBJECTIVES: At the end of this module the students are expected to:  Solve the unknown parts of a right triangle given some known parts;  Apply the concepts of trigonometric functions in problem situations where right triangles are involved;  Use the scientific calculator appropriately in finding the values of the unknowns; and  Solve different cases of problem situations involving angles of elevation and angles of depression. MECHNOLOGYC ANDALUYONG S OLLEGE OF T CIENCE & REVIEW: Find the exact value of the following: √𝟐 √𝟐 1.) 𝟐 5.) 𝟐 𝟏 2.) 𝟐 6.) 𝟏 𝟐 3.) √𝟑 7.) − √ 𝟐 𝟐 4.) √𝟑 8.) 𝟐 1 𝟐 MECHNOLOGYC ANDALUYONG OLLEGE OFS T CIENCE & SOLVING RIGHT TRIANGLES  For the right triangle shown below, we have MECHNOLOGYC ANDALUYONG OLLEGE OFS T CIENCE & SOLVING RIGHT TRIANGLES  Consider the right triangle ACB with acute angles A and B, and right angle C.  In naming a right triangle we use the three letters representing the 3 angles of the triangle, with the middle letter representing the right triangle. MECHNOLOGYC ANDALUYONG S OLLEGE OF T CIENCE & SOLVING RIGHT TRIANGLES  Solving a triangle means finding the measures of all the angles and sides of the triangle. If we are given the measures of two sides or the measures of a side and an acute angle of a right triangle, we can solve the triangle. MECHNOLOGYC ANDALUYONG S OLLEGE OF T CIENCE & SOLVING RIGHT TRIANGLES  In order to solve unknown parts of the right triangle, we can use Pythagorean theorem and the following definitions of the 𝑠𝑖𝑑𝑒 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝐴 𝑎 𝑠𝑖𝑑𝑒𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝐵 𝑏 sin trigonometric 𝐴= functions = of𝐵= sin and. = h𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 𝑐 h𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 𝑐 𝑠𝑖𝑑𝑒 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝐴 𝑏 𝑠𝑖𝑑𝑒 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝐵 𝑎 cos 𝐴= = cos 𝐵= = h𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 𝑐 h𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 𝑐 𝑠𝑖𝑑𝑒 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝐴 𝑎 𝑠𝑖𝑑𝑒𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝐵 𝑏 tan 𝐴= = tan 𝐵= = 𝑠𝑖𝑑𝑒 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝐴 𝑏 𝑠𝑖𝑑𝑒 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝐵 𝑎 MECHNOLOGYC ANDALUYONG S OLLEGE OF T CIENCE & SOLVING RIGHT TRIANGLES 𝑎 𝑏 sin 𝐴= =cos 𝐵 cos 𝐴= =sin 𝐵 𝑐 𝑐  Note that in triangle ACB, the Pythagorean Theorem says,.  Also, the sum of the acute angles is that is A + B = , A = - B and B = - A MECHNOLOGYC ANDALUYONG S OLLEGE OF T CIENCE & Examples: 1. Solve right triangle ACB with B = and c = 72.4cm. 58.20cm 𝟎 𝟓𝟑 𝟑𝟎 ′ 43.06cm MECHNOLOGYC ANDALUYONG S OLLEGE OF T CIENCE & Examples: 2. Find the unknown parts of right triangle ACB if A= and a= 24.07cm. 57.49cm 𝟎 𝟔𝟓 𝟏𝟓 ′ 52.21cm MECHNOLOGYC ANDALUYONG OLLEGE OF S T CIENCE & Examples: 3. Solve right triangle ACB if and. 𝟎 ′ 𝟒𝟐 𝟐 𝟎 𝟑𝟑. 𝟑𝟖 𝟎 ′ 𝟒𝟕 𝟑 𝟗 𝟐𝟔.𝟔𝟐 135.55cm MECHNOLOGYC ANDALUYONG S OLLEGE OF T CIENCE & APPLICATIONS OF RIGHT TRIANGLE  Applications of right triangles involve the angle of elevation and angle of depression. MECHNOLOGYC ANDALUYONG S OLLEGE OF T CIENCE & APPLICATIONS OF RIGHT TRIANGLE  The angle of elevation is the angle made by the horizontal axis by the line of sight from an observer to an object on a higher level and the angle of depression is the angle made by the horizontal axis by the line of sight of an observer to an object on a lower level. MECHNOLOGYC ANDALUYONG OLLEGE OF S T CIENCE & APPLICATIONS OF RIGHT TRIANGLE  If an observer at O looks at a point at P, then OP is called the line of sight, furthermore, if H is any point on a horizontal ray through O and in the same vertical plane as OP, then the angle from OH to OP is called the angle of elevation of P if P is above OH and is called the angle of depression of P if P is below OH. MECHNOLOGYC ANDALUYONG S OLLEGE OF T CIENCE & Examples: 1. Jay is flying a kite. The string of the kite makes an angle of with the ground. If the height of the kite is 21cm, find the length of the string that Jay has used. Ans: 31.08cm MECHNOLOGYC ANDALUYONG S OLLEGE OF T CIENCE & Examples: 2. A pole 60ft high casts a shadow 48ft long at a particular time. Find the angle of elevation of the sun at this instant. Ans: MECHNOLOGYC ANDALUYONG S OLLEGE OF T CIENCE & Examples: 3. From the top of a barn 25ft tall, Hannah saw a cat on the ground. The angle of depression of the cat is. How many feet must the cat walk to reach the barn? Ans: 29.74ft MECHNOLOGYC ANDALUYONG OLLEGE OFS T CIENCE & Practice Exercises: 1. Solve the right 2. Solve the right triangle in which c = triangle in which a= 37.2 and A =. 41.72 and b = 57.34. MECHNOLOGYC ANDALUYONG S OLLEGE OF T CIENCE & Practice Exercises: 3. Jun is walking along the straight road. He notices the top of a tower subtending an angle of with the ground at the point where he is standing. If the height of the tower is 35m, what is his horizontal distance from the tower.