5.1 Trigonometric Ratios of Acute Angles Unit 5
Label the sides of the following triangle relative to the angle theta.
θ
Complete the following formulas. Beside the formulas write the acronym we learned to help
remember them.
sin θ =
______
cos θ =
tan θ =
Example 1: In
D
DEF, find the length of DE, to the nearest metre.
32o
F
7m
E
You can also use Pythagorean Theorem to help solve a right angle triangle:



where c is always the HYPOTENUSE
can only be used to solve for side lengths, NOT angles
for right angle triangles ONLY
Reciprocal Trigonometric Ratios
Notice in Example 1 that the unknown side was in the denominator, which is more difficult to
solve than when the unknown is in the numerator. Using reciprocal trigonometric ratios, problems
like Example 1 can be solved with the unknown in the numerator.
The Reciprocal Ratios are cosecant (csc), secant (sec) and cotangent (cot).
They are defined as 1 divided by each of the primary trigonometric ratios:
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csc θ = 1 = hypotenuse
sin θ opposite
sec θ = 1 = hypotenuse
cos θ adjacent
cot θ = 1 = adjacent
tan θ opposite
Let's revisit Example 1 and solve it using a reciprocal ratio.
Example 2: In KLM, calculate cot θ and then determine θ to the nearest degree.
L
θ
3
M
2
K
Example 3: How long must a kite string be in order to fly 56m above the ground at an angle of 68 o?
Use a reciprocal trigonometric ratio to calculate your answer and round your answer to the
nearest metre.
Also, recall:
Angle of elevation:
the angle between the horizontal and the line of
sight when looking up at an object
Angle of depression
the angle between the horizontal and the line of
sight when looking down at an object
pg. 281 #5-6; (7-8)a; 9-11; 15
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