Trigonometry Tutorial & Problems

 

 

 

 

 

 

 

 

 

Characteristics of the Right Triangle

One of the three angles is 90º

The sum of the three angles is 180º.  (θ₁º+θ₂º+90º = 180º)

The two sides are length a and b

The long side, or hypotenuse, is length c

 

Pythagorean's Theorem:  In a right triangle whose legs have length a and b, and whose hypotenuse has length c, then c² = a² + b².

 

Example 1:  If side a = 5 inches and side b = 10 inches, what is the length of hypotenuse c?

            Pythagorean's Theorem states c² = a² + b²

                                                c² = (5 inches)² + (10 inches)²

                                                c² = 25 inches² + 100 inches²

                                                c² = 125 inches²

                                                c  = √(125 inches²)               This is an acceptable answer

                                                c  = √(5*5*5 inches²)

                                                c  = 5√5 inches                     This is the preferred answer

 

Example 2:  If side a = 6 inches and hypotenuse c = 12 inches, what is the length of side b?

            Pythagorean's Theorem states c² = a² + b²

                                                c² – a² = b²

                                                (12 inches)² – (6 inches)² = b²

                                                144 inches² – 36 inches² = b²

                                                108 inches² = b²

                                                √(108 inches²) = b               This is an acceptable answer

                                                √(2*2*3*3*3 inches²) = b

                                                6√3 inches = b                                  This is the preferred answer

 

 

 

 

 

 

 

 

 

 

Sine (abbreviated 'sin'):  In a right triangle, sin of an angle θ is the length of the opposite side divided by the length of the hypotenuse.

 

Example:  If side a = 3 inches, side b = 4 inches, and side c = 5 inches, what is sinθ₁ and sinθ₂?

 

sinθ₁ = (side a / hypotenuse c)

sinθ₁ = (3 inches / 5 inches)

sinθ₁ = 0.60

                       

sinθ₂ = (side b / hypotenuse c)

sinθ₂ = (4 inches / 5 inches)

sinθ₂ = 0.80

 

Cosine (abbreviated 'cos'):  In a right triangle, cos of an angle θ is the length of the adjacent side divided by the length of the hypotenuse.

 

cosθ₁ = (side b / hypotenuse c)

cosθ₁ = (4 inches / 5 inches)

cosθ₁ = 0.80

                       

cosθ₂ = (side b / hypotenuse c)

cosθ₂ = (3 inches / 5 inches)

cosθ₂ = 0.60

 

 

A common way to memorize these relationships is the term, ‘SOHCAHTOA’

            ‘SOH” represents ‘Sine is Opposite over Hypotenuse’

            ‘CAH’ represents ‘Cosine is Adjacent over Hypotenuse’

            ‘TOA’ represents ‘Tangent is Opposite over Adjacent’

 

					a
		b

 

 

 

 

 

 

 

 

 

 

Given the right triangle above and the following lengths, calculate the length of the missing side.  (Hint:  Don't forget the units)

 

 

Problem	Length of Side a	Length of Side b	Length of Hypotenuse c
1	3 meters	4 meters
2	5 yards	8 yards
3	1 inch	1 inch
4	8 miles	4 miles
5		1 foot	3 feet
6		8 mm	15 mm
7		2 yards	3 yards
8	5 feet		13 feet
9	8 inches		16 inches
10	1 mm		3 mm

 

 

 

 

 

 

 

 

 

 

 

Given the right triangle above, calculate the sin and cos given the following edge lengths.  The first one is done as an example.

 

 

#	Length a	Length b	Hypotenuse c	Problem a)	Problem b)
Ex	4 miles	6 miles	2√13 miles	sinθ1= _________ sinθ1 =  a/c =(4 miles) / (2√13 miles) =2/√13	cosθ1= __________ cosθ1 = b/c =(6 miles) / (2√13 miles) =3/√13
1	3 feet	4 feet	5 feet	sinθ1=	sinθ2=
2	5 meters	12 meters	13 meters	cosθ1=	sinθ1=
3	6 inches	4 inches	2√13 inches	cosθ1=	cosθ2=
4	1 mm	3√7 mm	8 mm	sinθ2=	cosθ2=
5	2√21 cm	4 cm	10 cm	sinθ1=	cosθ2=