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geometry special right triangles 6 x10
θ
Common Core Standard  G.SRT.6  High School Geometry
Follow this link for a trig function calculator to check your work

Easy calculations-Right Triangle Angle Calculator
right triangle cosine sin tangent
θ
special right triangles adjacent over opposite
θ
* Sine = Opposite/Hypotenuse
   Cosine = Adjacent/Hypotenuse
   Tangent = Opposite/Adjacent

* Trigonometry has been around from 1500 AD, and comes from the Greek words “trigonon” which means triangle and “metron” which means measure. 

* Right triangles have a 90 degree angle, and two additional complementary angles. 

* The longest side of a triangle that is found across from the right triangle symbol is the hypotenuse. The side that is across from the angle you are looking at is the “ opposite” side, and the remaining angle is the “adjacent” side.

* Tangent is the only trig function that doesn’t include hypotenuse.

* A popular acronym for remembering the trig functions is SOH CAH TOA, but other acronyms include: 

  • Oscar had A Heap Of Apples
  • O Heck Another Hour Of Algebra
  • Some Of Her Children Are Having Trouble Over Algebra
  • Some Old Hippie Came A Hopping Through Our Apartment

* The sine of an angle is equal to the cosine of the complementary angle.
For example, sin 40 = cos 50 and sin 10 = cos 80​

A TI-84 calculator and other calculators have sine, cosine, tangent, buttons that make calculating these values easy. Follow this link for directions on using a sine, cosine, tangent calculator.

* Cosine, sin, tangent are helpful in the real world whenever triangles are used. For example, bicycles frames, motorcycle frames, airplanes, roofing, framing of buildings, automobiles, boats all use triangles in their design.

* Sine divided by Cosine equals Tangent, Sin/Cos = Tan

Finding an angle

Given BC = 6 and AB = 10, find m<A.

1. Based on the given information choose a trig function. In this example, use sin because opposite and hypotenuse are provided.

      sin θ  = opposite/hypotenuse

2. When finding an angle use will use the inverse function of the calculator.  To calculate the Sin-1:

      Most calculators press "inv"

      On a TI calculator, press second, then Sin

​     Sin-1(6/10) =36.86 
     therefore, angle A = 36.86 degrees

3. Find the measure of angle B.

    The acute angles in a right triangle are     complementary therefore use,
    90 - ∠A

    90-36.86 = 53.14 ◦ angle B
Finding a side
Based on information provided, decide which trig function to use.  

Given:  BC = 6,  AC = 8 and θ=36.86 degrees 
​Find AB.

1.  You can use Sin because the opposite is given, and the hypotenuse is missing.

2.  Set up the ratio:  sin 36.86 = 6/x

3.  Use your calculator to take the 
      sin 36.86 =.599 

4. Plug in the .599 in the proportion.

      .599/1 = 6/x

5.  Cross multiply to find your hypotenuse (x) 
    .599x = 6
    x =6/.599 
    x =10 units is the length of the hypotenuse

Using trig functions to find missing angles of right triangles

Finding a side length of a right triangle using trig ratios

Opposite equals 6 units
Adjacent equals 8 units
Hypotenuse is unknown
​θ =36.86◦ 
Opposite = 6 units
Adjacent is unknown
Hypotenuse = 10 units
Transcript 


Welcome to MooMooMath. Today we are going to set up how to find sin and cosine. Let’s review, the Sin of an acute angle equals opposite over hypotenuse. The cosine of an acute angle equals adjacent over hypotenuse. How do we decide which side is which? First let’s look at angle A from angle a we have to look at our reference sides. If we go to our opposite side that will be our opposite side and I will label this o for opposite. The hypotenuse (labeled h) is always across from the fight angle so the adjacent or next to angle A. (labeled a) Now let’s add some measurements. Let’s say the opposite side is 6 and the hypotenuse is 10 in order to find my sin it will be the opposite over hypotenuse so the Sin of angle A will equal 6/10 which is my opposite over hypotenuse. The cosine is the adjacent over the hypotenuse I don’t know the length of a so I can use the Pythagorean Theorem to find this side. So I will take a squared plus b squared which is 6 plus c which is 10 squared so to write this out I get a^2+6^2 =10^2  which becomes a^2+36=100 so a^2 equals 64 so, A ,must be the length of 8 (take the square root of 64) I can now solve for A The Cos of A will be 8 over 10 because it is adjacent over hypotenuse and that is how I would solve. You can also look from angle B the opposite side would be the lower angle here, the adjacent is here and the hypotenuse is always opposite the right angle. So the opposite and adjacent will flip according to which angle. Hope this was helpful. 

Tangent,Sine,Cosine



Naming the sides of a right triangle

One key to using trig functions properly is identifying the sides of the right triangle correctly. 

Hypotenuse is always opposite the ninety degree angle

Opposite will be opposite the angle  ( Angle θ  in this example)

Adjacent will be next to angle
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The six trig functions

In addition to the primary functions there are three additional cofunctions

Cosecant =  hypotenuse/opposite

 or it can be written

1/sin

Secant = hypotenuse/adjacent

or it can be written

1/cosine

Cotangent = adjacent/opposite

or it can be written

1/tangent

 

The  cofunctions are reciprocals of sin,cosine,and tangent.

Cosecant is the reciprical of sin
If sin x = 5/13, then cosecant x = 13/5

Secant is the reciprocal of cosecant
If cosecant  x = 12/13, then secant x = 13/12

Cotangent is the reciprocal of tangent
If tanent x = 5/12, then cotangent  x = 12/5
* Sine, cosine, tangent are trigonometry functions that are used with right triangles to find side lengths and angles.