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Trigonometric Functions in Terms of Opp, Adj, Hyp. DISCUSSION Precalculus So far we have expressed the trigonometric functions in terms of a circle with radius r. (x, y) r y θ x In terms of a circle with radius r, our 6 trigonometric functions are: sin ______ ______ cos ______ ______ tan ______ ______ csc ______ ______ sec ______ ______ cot ______ ______ Notice that in the diagram above we also have a right triangle with legs x, y, and hypotenuse r. In terms of a right triangle we have another way of expressing our trigonometric functions that will still give the same values, but from a viewpoint of a right triangle rather than a circle. The right triangle is enlarged below: r y θ x With respect to the angle , the side with y can be considered to be the _________________ side, the side with x can be considered to be the ________________ side, and the radius is the ____________________. Our six trigonometric functions can now be expressed as: sin ________ ________ cos _______ _______ tan _________ _________ csc _________ _________ sec _________ _________ cot _________ _________ Notice that in a right triangle there are two angles that are not equal to 900. In the triangle below, these two angles are and . B α c a θ A b C With respect to angle θ the opposite side has length ______, the adjacent side has length ______, and the hypotenuse has length _____. The trigonometric functions for angle θ, then, are: sin ____________ ___ ____________ ___ csc ____________ ___ ____________ ___ cos ___________ ___ ___________ ___ sec ____________ ___ ____________ ___ tan _____________ ___ _____________ ___ cot ____________ ___ ____________ ___ With respect to angle α the opposite side has length ______, the adjacent side has length ______, and the hypotenuse has length _____. The trigonometric functions for angle θ, then, are: sin ____________ ___ ____________ ___ csc ____________ ___ ____________ ___ cos ___________ ___ ___________ ___ sec ____________ ___ ____________ ___ tan _____________ ___ _____________ ___ cot ____________ ___ ____________ ___ Notice that the two acute angles in a right triangle are _______________ of one another (meaning they add up to ________). Also notice that for complementary angles the sin of one angle is equal to the _________ of its complement. We have seen this already in looking at the trigonometric function values for two of our special angles, 300 and 600 (which are complements of one another). The sin 300 = _______ and the cos 600 = _______. Also, the sin 600 = _______ and the cos 300 = ____. Try this example: For angle B: Opposite side = ______ B Adjacent side = ________ 5 3 Hypotenuse = _________ A 4 sin B _____ _____ cos B _____ _____ tan B _____ _____ For angle A: Opp side = _____ Adj side = ______ Hyp = ______ sin A _____ _____ cos A _____ _____ tan A _____ _____ If one of the acute angles of a right triangle is known and one side length is known, trig functions can be used to determine the value for any other side length. Try these examples: opposite adjacent opposite sin cos tan hypotenuse hypotenuse adjacent Which trig function will be used? ____________ Solve for the unknown length. 30 7 ______ 30 0 ______ ______ g g ______ ______ ______ c 75 50 Which trig function will be used? __________ Solve for the unknown length. ______ ______ c ______ ______ ______ c ______ ______ ______ 75 0 Trigonometric functions are used in many real world problems where a situation can be expressed in terms of a right triangle: Example Word Problems sin opposite hypotenuse cos adjacent hypotenuse tan opposite adjacent A plane rises at an angle of 10o with respect to the ground. Find the height, h, above the ground the plane is after it has traveled 200 meters on its path. (1.) Draw a sketch of the situation. Label the angle, known length, and unknown length symbol on the diagram. (2.) Solve for the unknown height. sin opposite hypotenuse cos adjacent hypotenuse tan opposite adjacent A board leans against a building and makes an angle of 40o with the ground. If the end of the board is 4 meters from the building, what is the length, l, of the board? (1.) Draw a sketch of the situation. Label the angle, known length, and unknown length symbol on the diagram. (2.) Solve for the unknown length. Inverse trigonometric functions can also be used to find an angle in a word problem if two sides of a right triangle are known and you want to determine an angle. Example: You are standing 100 meters away from a building that is 170 meters tall. At what angle must you look up at to see the top of the building? (1.) Draw a sketch of the situation. Label the angle, known length, and unknown length symbol on the diagram. (2.) Solve for the unknown angle. Trigonometric Functions in terms of Opp., Adj, and Hyp. Precalculus For each triangle, find the indicated trig function value. Express answers as fractions in simplest form. sin opposite hypotenuse cos adjacent hypotenuse tan opposite adjacent Hypotenuse Opposite leg θ Adjacent leg B 1.) 61 11 C 60 A Sin A = ______ ______ Sin B = ______ ______ Cos A = ______ ______ Cos B = ______ ______ Tan A = ______ ______ Tan B = ______ ______ B 2.) 58 A 40 42 C Tan A = ______ ______ Sin A = ______ ______ Cos B = Sin B = ______ ______ ______ ______ Tan B = Cos A = ______ ______ ______ ______ Hypotenuse Opposite leg θ Adjacent leg Fill in the blanks and determine the unknown side length using trigonometry. sin opposite hypotenuse cos adjacent hypotenuse tan opposite adjacent 3.) Which trig function will be used? ____________ Solve for the unknown length. 50 3 ______ 50 0 g ______ ______ g ______ ______ ______ 4.) d 28 5 Which trig function will be used? __________ Solve for the unknown length. ______ ______ d ______ ______ ______ ______ 28 0 5.) 4 75 s Which trig function will be used? ____________ Solve for the unknown length. 6.) 58 h 8 Which trig function will be used? ____________ Solve for the unknown length. Applications: sin opposite hypotenuse cos adjacent hypotenuse tan opposite adjacent 7.) Someone is flying a kite. The person lets out 40 meters of string. The angle the string makes with the ground is 300. How high, h, is the kite in the air? (1.) Draw a sketch of the situation. Label the angle, known length, and unknown length symbol on the diagram. (2.) Solve for the unknown length. 8.) A ladder makes an angle of 600 with the ground. Its top extends 5 m up the side of a building. What is the length, l, of the ladder? (1.) Draw a sketch of the situation. Label the angle, known length, and unknown length symbol on the diagram. (2.) Solve for the unknown length. sin opposite hypotenuse cos adjacent hypotenuse tan opposite adjacent 9.) A street goes up at an angle of 50. What will the horizontal distance, d, be between two points on the road that differ from one another by 10 meters in height? (1.) Draw a sketch of the situation. Label the angle, known length, and unknown length symbol on the diagram. (2.) Solve for the unknown distance. 10.) The stairway in a home goes up at an angle of 360. What is the length, l, of the stairway if the height gain in going up the stairs is 3 meters? (1.) Draw a sketch of the situation. Label the angle, known length, and unknown length symbol on the diagram. (2.) Solve for the unknown stairway length, l. opposite adjacent opposite cos tan hypotenuse hypotenuse adjacent 11.) The wheelchair ramp going into a home is at an angle of 80 with the ground. In total, the vertical height rise of the ramp is 1.3 m. Horizontally, what is the distance, d, from the home where the ramp starts? (1.) Draw a sketch of the situation. Label the angle, known height, and unknown distance symbol on the diagram. sin (2.) Solve for the unknown distance, d. 12.) A surveyor wants to determine the height of a vertical cliff without actually climbing it. She walks 85 meters away from the bottom of the cliff and measures the angle between the ground and the top of the cliff. If this angle is 52o, how high, h, is the cliff? (1.) Draw a sketch of the situation. Label the angle, known height, and unknown distance symbol on the diagram. (2.) Solve for the unknown height, h. 13.) At a point on the ground 4000 meters from where a plane has lifted off the plane is 300 meters in the air. At what angle in degrees is the plane rising? (1.) Draw a sketch of the situation. (2.) Calculate the angle in degrees the plane is rising at.