# Trigonometric Functions of Acute Angels

by
**Mialonnet**

Last updated 6 years ago

** Discipline: **

Math ** Subject: **

Pre-Calculus
** Grade:**

11

90°

0°

REMEMBER :

A single acute angle θ of a right triangle determines six distinct ratios of side lengths. EACH ratio can be considered a function or θ(THETA) as θ takes on values from 0° to 90° OR 0 radians to π/2 radians.

Adjacent

Opposite

Sine: sinθ = opp/hypCosine: cosθ = adj/hyp Tangent: tanθ = opp/adj__________________________Cosecant: cscθ = 1/sine (reciporical of sine) Secant: secθ = 1/cos (reciporical of cosine) cotangent: cotθ = 1/tan (recipoical of tangent)

Hypotenuse

(π/2)

4.2: Trigonometric Functions of Acute Angles

ff

Tip: Whenever your on a problem and trying to find the reciporical, flip the answer. But if extra work is needed, do it!

SOH | CAH | TOAi p y o d y a p dn p p s j p n p j

Side ratios for some angles that appear in right triangles can be found geometrically.

2 types of acute triangles :

EXAMPLE 1: 45-45-90° TriangleFind the values of all six trigonometric functions for an angle of 45. Assume the legnths of the two legs are 1.

1. 45-45-90°Triangle 2. 30-60-90° Triangle

First step is to do the Pythagorean Theroem: 1^2+ 1^2 = c^2 1+1=c^2 2=c^2 √2=c

45°

1

1

So now you can plug in the missing side and find the six functions.

√2

Sin45°: (opp/hyp) 1/√2 = √2/2cos45°: (adj/hyp) 1/√2 = √2/2tan45°: (opp/adj)1/1 = 1

csc45°: (hyp/opp)√2/1 = √2sec45°: (hyp/adj) √2/1 = √2cot45°: (adj/opp) 1/1 = 1

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