Trigonometric Functions on Right Triangles - Video Tutorials & Practice Problems
Introduction to Trigonometric Functions
Given the right triangle below, evaluate cosθ.
cosθ=178
cosθ=158
cosθ=1715
cosθ=815
Given the right triangle below, evaluate tanθ.
tanθ=53
tanθ=54
tanθ=34
tanθ=43
Fundamental Trigonometric Identities
If tanθ=512, find the values of the five other trigonometric functions. Rationalize the denominators if necessary.
sinθ=1312,cosθ=135,cotθ=125,secθ=513,cscθ=1213
sinθ=135,cosθ=1312,cotθ=125,secθ=1213,cscθ=513
sinθ=1312,cosθ=135,cotθ=−125,secθ=−513,cscθ=−1213
sinθ=135,cosθ=1312,cotθ=−125,secθ=−1213,cscθ=−513
If sinθ=1717, find the values of the five other trigonometric functions. Rationalize the denominators if necessary.
cosθ=417,tanθ=41,cotθ=4,secθ=17,cscθ=17417
cosθ=417,tanθ=−41,cotθ=−4,secθ=17,cscθ=17417
cosθ=17417,tanθ=−41,cotθ=−4,secθ=417,cscθ=17
cosθ=17417,tanθ=41,cotθ=4,secθ=417,cscθ=17
Introduction to Inverse Trig Functions
Given the right triangle below, use the sine function to write a trigonometric expression for the missing angle θ.
θ=sin−1(135)
θ=sin−1(1312)
θ=sin−1(125)
θ=sin−1(1213)
Given the right triangle below, use the cosine function to write a trigonometric expression for the missing angle ϕ.
ϕ=cos−1(1312)
ϕ=cos−1(513)
ϕ=cos−1(1213)
ϕ=cos−1(135)
How to Use a Calculator for Trig Functions
What is a positive value of A in the interval [0°,90°) that will make the following statement true? Express the answer in four decimal places.
sinA=0.9235
22.5568°
67.4432°
22.4432°
33.5438°
What is the positive value of P in the interval [0°,90°) that will make the following statement true? Express the answer in four decimal places.
cotP=5.2371
55.8102°
34.1898°
10.8102°
79.1898°
What is the positive value of D in the interval [0,2π) that will make the following statement true? Express the answer in four decimal places.
secD=3.2842
0.3094 rad
1.2614 rad
0.4760 rad
1.0934 rad
Example 1
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- CONCEPT PREVIEW Match the measure of bearing in Column I with the appropriate graph in Column II. I. ...
- CONCEPT PREVIEW Fill in the blank(s) to correctly complete each sentence. Given tan θ = 1/cot θ , two equi...
- CONCEPT PREVIEW Match the measure of bearing in Column I with the appropriate graph in Column II. I. ...
- CONCEPT PREVIEW Determine whether each statement is possible or impossible. sin θ = 1/2 , csc θ = 2
- CONCEPT PREVIEW Match the measure of bearing in Column I with the appropriate graph in Column II. I. ...
- CONCEPT PREVIEW Determine whether each statement is possible or impossible. cos θ = 1.5
- CONCEPT PREVIEW Match the measure of bearing in Column I with the appropriate graph in Column II. I. ...
- CONCEPT PREVIEW Determine whether each statement is possible or impossible. sin² θ + cos² θ = 2
- Concept Check The two methods of expressing bearing can be interpreted using a rectangular coordinate system. ...
- Use the figure shown to solve Exercises 13–16. Find the bearing from O to A.
- Use the appropriate reciprocal identity to find each function value. Rationalize denominators when applicable....
- Concept Check The two methods of expressing bearing can be interpreted using a rectangular coordinate system. ...
- Concept Check The two methods of expressing bearing can be interpreted using a rectangular coordinate system. ...
- Use the appropriate reciprocal identity to find each function value. Rationalize denominators when applicable....
- Concept Check The two methods of expressing bearing can be interpreted using a rectangular coordinate system. ...
- Solve each problem. See Examples 1 and 2. Distance Traveled by a Ship A ship travels 55 km on a bearing of 27...
- Use the appropriate reciprocal identity to find each function value. Rationalize denominators when applicable....
- Solve each problem. See Examples 1 and 2. Distance between Two Ships Two ships leave a port at the same time....
- Use the appropriate reciprocal identity to find each function value. Rationalize denominators when applicable....
- Solve each problem. See Examples 1 and 2. Distance between Two Ships A ship leaves its home port and sails on...
- Concept Check What is wrong with the following item that appears on a trigonometry test? "Find sec θ , giv...
- Determine the signs of the trigonometric functions of an angle in standard position with the given measure. Se...
- Solve each problem. See Examples 1 and 2. Distance between Two Cities The bearing from Atlanta to Macon is S ...
- Determine the signs of the trigonometric functions of an angle in standard position with the given measure. Se...
- Solve each problem. See Examples 3 and 4. Height of an Antenna A scanner antenna is on top of the center of a...
- Determine the signs of the trigonometric functions of an angle in standard position with the given measure. Se...
- Determine the signs of the trigonometric functions of an angle in standard position with the given measure. Se...
- Solve each problem. See Examples 3 and 4. Distance through a Tunnel A tunnel is to be built from point A to...
- Give all six trigonometric function values for each angle θ . Rationalize denominators when applicable. co...
- Identify the quadrant (or possible quadrants) of an angle θ that satisfies the given conditions. See Example ...
- Give all six trigonometric function values for each angle θ . Rationalize denominators when applicable. se...
- Give all six trigonometric function values for each angle θ . Rationalize denominators when applicable. se...
- Identify the quadrant (or possible quadrants) of an angle θ that satisfies the given conditions. See Example ...
- Determine whether each statement is possible or impossible. a. sec θ = ―2/3
- Determine whether each statement is possible or impossible. b. tan θ = 1.4
- Determine whether each statement is possible or impossible. c. cos θ = 5
- Solve each problem. See Examples 3 and 4. The figure to the right indicates that the equation of a line passi...
- Identify the quadrant (or possible quadrants) of an angle θ that satisfies the given conditions. See Example ...
- Identify the quadrant (or possible quadrants) of an angle θ that satisfies the given conditions. See Example ...
- Solve each problem. Height of a Lunar Peak The lunar mountain peak Huygens has a height of 21,000 ft. The s...
- Solve each problem. (Source for Exercises 49 and 50: Parker, M., Editor, She Does Math, Mathematical Associati...
- Determine whether each statement is possible or impossible. See Example 4. sin θ = 3
- Determine whether each statement is possible or impossible. See Example 4. tan θ = 0.93
- Solve each problem. (Source for Exercises 49 and 50: Parker, M., Editor, She Does Math, Mathematical Associati...
- Solve each problem. (Source for Exercises 49 and 50: Parker, M., Editor, She Does Math, Mathematical Associati...
- In Exercises 61–62, use the figures shown to find the bearing from O to A.
- Determine whether each statement is possible or impossible. See Example 4. csc θ = 100
- Determine whether each statement is possible or impossible. See Example 4. cot θ = ―6
- Use identities to solve each of the following. Rationalize denominators when applicable. See Examples 5–7. ...
- Use identities to solve each of the following. Rationalize denominators when applicable. See Examples 5–7. ...
- If θ is an acute angle and cos θ = 1/3, find csc (𝜋/2 - θ).
- Give all six trigonometric function values for each angle θ. Rationalize denominators when applicable. See Ex...
- Give all six trigonometric function values for each angle θ. Rationalize denominators when applicable. See Ex...
- Give all six trigonometric function values for each angle θ. Rationalize denominators when applicable. See Ex...
- Give all six trigonometric function values for each angle θ. Rationalize denominators when applicable. See Ex...
- Give all six trigonometric function values for each angle θ. Rationalize denominators when applicable. See Ex...
- Concept Check Suppose that 90° < θ < 180° . Find the sign of each function value. tan θ/2
- Concept Check Suppose that 90° < θ < 180° . Find the sign of each function value. cot (θ + 180°)
- Concept Check Suppose that 90° < θ < 180° . Find the sign of each function value. cos ( ―θ)
- Concept Check Suppose that ―90° < θ < 90° . Find the sign of each function value. sec θ/2
- Concept Check Suppose that ―90° < θ < 90° . Find the sign of each function value. sec(―θ)
- Concept Check Suppose that ―90° < θ < 90° . Find the sign of each function value. cos(θ―180°)
- Concept Check Find a solution for each equation. tan (3θ ― 4°) = 1 / [cot(5θ ― 8°)]
- Concept Check Find a solution for each equation. sin(4θ + 2°) csc(3θ + 5°) = 1
- Concept Check Find a solution for each equation. sec(2θ + 6°) cos(5θ + 3°) = 1