Distinguishing Patterns by Relationships
Applying Trigonometric Ratios
MINDS ON
In the previous activity, you looked at three trigonometric ratios that use two sides of a right angle triangle along with one of the acute angles. The word trigonometry comes from Greek, where “trigon” refers to “triangles” and “metry “means “a measure”, so that the word “trigonometry” literally means “the measuring (of the angles and sides) of a triangle. The Pythagorean Theorem is another relation that is associated with a right angle triangle, but it uses all three sides of the right triangle and does not include an angle in the relation.
You have learned about and used the Pythagorean Theorem and the relationship between the side lengths in previous courses. To review the Pythagorean Theorem watch the presentation that can be found at mathisfun.com.
Trigonometry
Using Image 1 (below) as a model, create a chart for trigonometry (Trigonometry describes the mathematical relationships involving lengths and angles of triangles) that will be used as you continue to learn how to use the relationships to solve problems.
The relationships include:
- the 3 trigonometric relations - sine, cosine, and tangent of an angle;
- the Pythagorean Theorem.
You can list each relation more than once in the chart.
Name your document Working with Triangles.
The Pythagorean Theorem has been entered in the first row for the case where the hypotenuse is the unknown value.
As you continue through this activity, consider other variations where the Pythagorean Theorem can be used to solve for an unknown value.
ACTION
Mathematical Processes
In this activity, the Mathematical Process Selecting Tools And Computational Strategies is the focus.
Open your document U2 Mathematical Processes and read the description for the process.
As you complete the activity, notice when you are considering the information given before you choose a strategy and insert your record below the description of the process.
Save to your Portfolio.
Names of Sides and a Trigonometric Ratio
In Activity 3, you defined three ratios using the labels, opposite, adjacent and hypotenuse.
For each triangle below, identify which ratio has correctly matched the sides to the ratio:
For △CDE above, which ratio has correctly matched the sides to the ratio:
a) cos C = CE/ED
b) tan C = ED/CD
c) sin C = CD/EC
d) cos C = DE/EC
tan C = ED/CD
For △KLM above, which ratio has correctly matched the sides to the ratio:
a) tan L = MK/LK
b) cos L = LK/LM
c) sin L = MK/LK
d) tan L = LM/MK
c) sin L = MK/LK
For △RST above, which ratio has correctly matched the sides to the ratio:
a) sin S = TR/SR
b) cos S = ST/SR
c) tan S = SR/RT
d) cos S = SR/ST
d) cos S = SR/ST
For each triangle below, identify the correct ratio for the given triangle and stated side lengths.
For △XYZ above, identify the correct ratio for the given triangle and stated side lengths:
a) cos X = 12/35
b) tan X = 12/37
c) sin X = 35/37
d) tan X = 12/35
c) sin X = 35/37
For △ABD above, identify the correct ratio for the given triangle and stated side lengths:
a) sin B = 15/17
b) tan B = 17/8
c) sin B = 8/15
d) cos B = 15/17
d) cos B = 15/17
△DEF above, identify the correct ratio for the given triangle and stated side lengths:
a) tan D = 12/9
b) sin D=9/15
c) cos D = 12/15
d) tan D = 9/12
a) tan D = 12/9
You can further test your understanding of the trigonometric ratios using this resource.
If you had to try again on 2 or more of the questions on the previous quiz, you may want to watch the following video about labelling sides in a right triangle and the trigonometric ratios:
Using Trigonometric Ratios or the Pythagorean Theorem to Determine Values
In mathematics, formulas are developed because patterns have been observed that occur over and over again. When a formula is developed, it means that you do not have to work out the relation each time. You can use the formula. And a particular formula is chosen because there is a value that we need to know.
Trigonometric ratios (formulas) were developed thousands of years ago as a way to calculate distances; people were able to chart stars and navigate seas. In these types of problems, large distances were represented as side lengths of scaled-down triangles, i.e., similar triangles. When some angles or sides in a triangle can be measured, then the other sides and angles can be determined. Similarly, the Pythagorean Theorem was developed thousands of years ago and is a relation between the three sides of a right triangle.
Many problems in real life can be represented using triangles - the concept is used in surveying, astronomy, architecture, construction and many other fields. When a situation can be modelled using a right triangle but not all angles and side lengths are known, you can use a trigonometric ratio or the Pythagorean Theorem to determine the unknown value.
Determining Values
In this section, you will
- Assess the information (angles and side lengths) provided in the diagram of the triangle.
- Decide whether you need to determine an unknown angle or side length.
- Select the appropriate trigonometric ratio or the Pythagorean Theorem to combine the given values with the unknown value.
- Write the appropriate formula, substituting values for the variables in the formula. This will be called the “solution set-up”.
Open U2A4 Examples to record your steps as you follow the examples.
Set up the appropriate formula to determine the unknown values (shown with a ?).
You are not required to finish solving for the unknown value.
Save your work to your Portfolio.
Example 1: Using Trig Ratios
example1
Example 2: Using Trig Ratios
example2
Example 3: Using Trig Ratios
example3
Example 4: Using Trig Ratios
example4
Example 5: Using Trig Ratios
Real World Calculations
Recall the task you completed in the previous unit where you determined the height of a tree by measuring shadows. Measuring devices are available to measure angles so the tree height could have been determined using trigonometric ratios.
In this situation, the distance from the tree was measured and the angle from the ground to the top of the tree was determined using an angle-measuring device. Notice that the shape we see is a triangle.
To model this situation, you want to transfer the measured information onto a diagram of a triangle.
One assumption we will make is that the tree is at a right angle to the ground. Now the same steps that were followed before can be used.
example5
Complete the similar questions in the right column and on the last page. Check the right column with the solution page U2A4 Example Solutions.
Trigonometry
Download and complete the worsheet U2A4 Trigonometry 1.
Working with Triangles
Open your Working with Triangles document from the beginning of the activity.
Continue to make entries for the Pythagorean Theorem and the primary trigonometric ratios.
Think about the values you may need to determine and what you need to know to do this.
CONSOLIDATION
Mathematical Processes
In this activity, the Mathematical Processes Selecting Tools And Computational Strategies was the focus.
Open your document U2 Mathematical Processes and complete your record of when you noticed that you considered the given information before choosing a strategy.
Save to your Portfolio.
Reflection
Open your document Unit 2 Reflections.
Reflect on the new ideas, vocabulary, and trigonometric ratios that you learned about in this activity.
Choose one of the sentence stems in the document to start your reflection.
Save to your Portfolio.
Congratulations, you have completed Unit 2, Activity 4. You may move on to Unit 2, Activity 5.
Learning Goals and Success Criteria
Learning Skills and Work Habits