Created by Jindriska Liskova , 2014 http://www.teacherspayteachers.com/Store/Jindriska
Theory Trigonometric functions relate the angles of a right triangle to the lengths of its sides . The most familiar functions are sine (sin) , cosine (cos) , tangent (tan) which are defined as ratios of two sides of a right triangle (see the pictures below) . ‹ Firstly, we should choose the angle we are interested in and name the three sides of the triangle . For the angle A, the sides of the triangle are named as follows:
The hypotenuse is the side opposite the right angle, in this case side c . The hypotenuse is always the longest side of a right-angled triangle .
TheTheThe opposite side is the side opposite to the angle we are interested in (angle A), in this case side aaa. opposite opposite opposite opposite TheTheThe adjacent side is the side which is adjacent to the angle of interest (angle A) and is not apposite the right angle, in this case side bbb.
adjacent
‹ Then the trigonometric functions are defined as follows:
Sin ∝ === coscoscos ∝ === tantantan ∝ ===
Sin ∝ = coscoscos ∝ = tantantan ∝ =
α
Aplication: Computing unknown lengths and angles in right triangles .
Example 111 (solve the unknown side) How high is the plane in the air if it has flown 42 km while going upward at the angle of 8°? Solution: Step 1 We are interested in the 8° angle (as it is the only one given) therefore we label the sides of the triangle as follows:
x= ?
opposite 42 km xxx = ? 8°8°8° Step 2 42 km We choose the right trigonometric function according
to these two sides: - the side we need to solve (in this case, the opposite ) 8°8°8° - the side we know the length of (in this case, the hypotenuse ) sine
Step 3 Solution: Sin 8° = = x = 42 ∙ sin 8° = 5,8 km Example 2 (solve the unknown angle) At what angle is the skier skiing down the slope?
Solution:
Step 1
We are interested in angle x°x°x°,x° so we name the sides of the triangle as follows:
x= ? 120120120 mmm
opposite xxx
adjacent 448 mmm 120 m Step 2 We choose the right trigonometric function according xxx = ? to the two sides we know - in this case we know the opposite and adjacent xxx
448 mmm tangent Step 3 Solution: tan x = = x = tan -1 = 11151555°°°° tan-1 = inverse function, we use it to calculate an angle. Name: Class: Date:
Task 111 Determine the height of the Eiffel Tower, knowing that at 600m away and you can see the top at an angle of 28° 22´.
xxx = ?
282828 ° 222222
600 m Name: Class: Date:
Task 2 How long is the kite´s string if the kite is flying 10 meters high at an angle of 65°?
xxx = ?
1110 m
65° Name: Class: Date:
Task 3 How long of a shadow does a 175cm tall person cast with the sun 60° in the sky?
175 cm
60°
xxx === ??? Name: Class: Date:
Task 4 How far does the bird have to fly to reach the apple in the tree?
xxx = ?
40° 6 m Name: Class: Date:
Task 5 At what height is the ladder touching the wall if the ladder is 2.5 meters long and leaning at an angle of 30°?
x = ? xx = ?
222...5 m
30° Name: Class: Date:
Task 6 What angle does the rocket have to be launched at to reach the rocket base 300 km above Earth?
300 km
xxx = ?
xxx
90 km Name: Class: Date:
Task 7 What angle does the arrow have to be shot at to hit the bullseye?
xxx = ? 38 m
xxx
24 m
Name: Class: Date:
Task 8 At what angle is the cyclist riding on the steep road?
CÍL
1200 m 50 m
xxx
xxx = ?
Name: Class: Date:
Task 9 What is the angle of the cableway leading up to the mountain?
210 m xxx = ?
xxx
Name: Class: Date:
Task 10 What is the angle of the dolphin diving into the sea?
20 m
xxx = ?
xxx
9 m Name: Class: Date:
Answer keykey::::
Trig. function Answer Task 1 tangent 324m Task 2 sine 11m Task 3 tangent 101cm Task 4 cosine 7.8m Task 5 sine 1.25m Task 6 tangent 73° Task 7 cosine 51° Task 8 sine 2° Task 9 sine 25° Task 10 cosine 63° Thank you for downloading this product!
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© 2014 Jindriska Liskova