# Common Core: High School - Geometry : Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles

## Example Questions

### Example Question #4 : Understand That By Similarity, Side Ratios In Right Triangles Are Properties Of The Angles In The Triangle, Leading To Definitions Of Trigonometric Ratios For Acute Angles

Using the sides of a right triangle, what is the definition of ?

Explanation:

We can see this to be true by looking at the following triangle.

We are considering .  The opposite side to  is  has two adjacent sides, but since  is opposite to the right angle, this is the hypotenuse of the triangle.  So the adjacent side to  must be .  We can set up the following equation to check and make sure that .

This shows that .

### Example Question #1 : Understand That By Similarity, Side Ratios In Right Triangles Are Properties Of The Angles In The Triangle, Leading To Definitions Of Trigonometric Ratios For Acute Angles

True or False: By similarity, side ratios in right triangles are properties of the angles in the triangle.

False

True

True

Explanation:

Similar right triangles are two right triangles that differ in side lengths but have congruent corresponding angles.  This means that if you have an angle, , in the first triangle and an angle, , in the second triangle.  So  .If we are considering the cosine of these two angles.

Side ratios would also follow from computing the sine and tangent of the angles using their sides as well.  This shows that the side ratios are properties of the angles in the triangles.

### Example Question #1 : Understand That By Similarity, Side Ratios In Right Triangles Are Properties Of The Angles In The Triangle, Leading To Definitions Of Trigonometric Ratios For Acute Angles

Assume the two triangles below are similar.  Using the fact that their sides ratios are , what trigonometric function could this represent for angles and

,

Explanation:

We must begin by manipulating our equation of the side ratios to get fractions that include both of the sides of the same triangles:

multiply both sides by

divide both sides by

If we look at angle  we see that  is the .  Looking at angle we see that  is also .  This is the definition of the cosine function of an angle. So:

### Example Question #101 : Similarity, Right Triangles, & Trigonometry

Consider the following right triangle.  Use trigonometric ratios to solve for  and .

Explanation:

We are given angle  and we need to find sides  and .  So first we need to think about what relation these sides are to angle .  Side  is the hypotenuse of the triangle since it is the opposite of the right angle.  Side  is opposite angle .  Recall that the trigonometric ratio that corresponds to sine is .  We can solve for the missing sides by solving for the following equation.

It follows that:

### Example Question #1 : Understand That By Similarity, Side Ratios In Right Triangles Are Properties Of The Angles In The Triangle, Leading To Definitions Of Trigonometric Ratios For Acute Angles

Using the information from the triangle below, use the side ratios to find .

Explanation:

The definition of the sine of an angle is .  So here we must determine which side is opposite of angle  and which side is the hypotenuse of this triangle.  We know that side  is the hypotenuse since it is opposite of the right angle.  Side  is directly opposite of angle .  So:

### Example Question #1 : Understand That By Similarity, Side Ratios In Right Triangles Are Properties Of The Angles In The Triangle, Leading To Definitions Of Trigonometric Ratios For Acute Angles

Consider the following right triangle.  Use trigonometric ratios to solve for sides  and .

Explanation:

We are given angle  and we need to find sides  and .  So first we need to think about what relation these sides are to angle .  Side  is the hypotenuse of the triangle since it is opposite of the right angle.  So side must be the side adjacent to angle .  Recall that the trigonometric ratio that corresponds to cosine is .  We can solve for the missing sides by solving for the following equation.

It follows that:

### Example Question #1 : Understand That By Similarity, Side Ratios In Right Triangles Are Properties Of The Angles In The Triangle, Leading To Definitions Of Trigonometric Ratios For Acute Angles

Using the information of the side lengths in the triangle below, use the side ratios to find the .

Explanation:

The definition of cosine of an acute angle is .  So here we must determine what sides are adjacent and which is the hypotenuse.  The hypotenuse is the easiest to pick out.  This is the side that is directly across from the right angle in a right triangle.  Our hypotenuse is .  Now we must choose the adjacent side.  Adjacent means next to, and since  is our hypotenuse then  must be our adjacent side.  So:

### Example Question #1 : Understand That By Similarity, Side Ratios In Right Triangles Are Properties Of The Angles In The Triangle, Leading To Definitions Of Trigonometric Ratios For Acute Angles

Find angle  using trigonometric ratios.

60 degrees

75 degrees

50 degrees

45 degrees

45 degrees

Explanation:

We are given the adjacent side to angle  and the hypotenuse of the triangle.  We can use this to set up the trigonometric ratio  which we know to be the definition of cosine.  We can solve for angle  using the following equation.

### Example Question #5 : Understand That By Similarity, Side Ratios In Right Triangles Are Properties Of The Angles In The Triangle, Leading To Definitions Of Trigonometric Ratios For Acute Angles

Assume the two triangles below are similar.  Using the fact that their side ratios are , what trigonomic function could this represent for angles  and .

Explanation:

We must begin by manipulating our equation of the side ratios to get fractions that include both of the sides of the same triangles:

multiply both sides by

divide both sides by

If we look at angle  we see that  is the .  Looking at angle  we see that  is also the .  This is the definition of the tangent of an angle.  So:

### Example Question #101 : Similarity, Right Triangles, & Trigonometry

Assume the following two triangles are similar.  Using the fact that their side ratios are , what trigonomic function could this represent for angles  and ?

Explanation:

We must begin by manipulating our equation of the side ratios to get fractions that include both of the sides of the same triangles:

If we look at angle , we see that  is the .  Looking at angle  we see that  is also the .  This is the definition of the sine function.  So: