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Exercises: Understand Trigonometric Ratios

Show your work for each problem. Round decimal answers to the nearest hundredth unless otherwise stated.

Grade 9·22 problems·~30 min·Common Core Math - HS Geometry·standard·hsg-srt-c-6
Work through problems with immediate feedback
A

Warm-Up: Review What You Know

These problems review skills you have already learned.

1.

Two triangles both have a 90° angle and a 35° angle. Which criterion guarantees they are similar?

2.

Triangle PQRPQR is similar to triangle STUSTU. Side PQ=6PQ = 6, side ST=9ST = 9, and side QR=8QR = 8. What is side TUTU?

3.

Right triangle ABCABC has legs AC=5AC = 5 and BC=12BC = 12 and hypotenuse AB=13AB = 13. Right triangle DEFDEF is similar to △ABC\triangle ABC with hypotenuse DE=26DE = 26. What is the ratio EFDE\frac{EF}{DE} in triangle DEFDEF? Express as a fraction.

B

Fluency Practice

Calculate the trigonometric ratios. Express answers as fractions or decimals rounded to the nearest hundredth.

Right triangle ABC with sides labeled 3, 4, and 5
1.

In right triangle ABCABC with a right angle at CC, BC=3BC = 3, AC=4AC = 4, and AB=5AB = 5. What is sin⁡(A)\sin(A)? Express as a fraction.

2.

Using the same triangle (sides 3, 4, 5 with right angle at CC), what is cos⁡(A)\cos(A)? Express as a fraction.

Right triangle with sides 5, 12, 13 and angle X at the bottom-left vertex
3.

Right triangle XYZXYZ has a right angle at ZZ, with YZ=5YZ = 5, XZ=12XZ = 12, and XY=13XY = 13. Which expression correctly gives tan⁡(X)\tan(X)?

4.

Right triangle PQRPQR has a right angle at RR, with QR=9QR = 9 and PR=4PR = 4. What is tan⁡(P)\tan(P)? Express as a fraction.

5.

In right triangle ABCABC with right angle at CC, angle A=40°A = 40\degree and AB=15AB = 15. Using sin⁡(40°)≈0.643\sin(40\degree) \approx 0.643, find the length of side BCBC. Round to the nearest hundredth.

C

Mixed Practice

These problems test the same skills in different ways.

Two right triangles with 40° angles of different sizes, both showing opposite/hypotenuse = 0.643
1.

Why is sin⁡(40°)\sin(40\degree) the same value in every right triangle that contains a 40° angle, regardless of the triangle's size?

2.

Right triangle MNPMNP has a right angle at PP. For angle MM: the side opposite is NPNP, the side adjacent is   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲   , and the hypotenuse is   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲   . Therefore cos⁡(M)=000000/000000\cos(M) = {\hspace{0.2em}\fbox{\phantom{000000}}\hspace{0.2em}}/{\hspace{0.2em}\fbox{\phantom{000000}}\hspace{0.2em}}.

adjacent side name:
hypotenuse name:
numerator of cos(M):
denominator of cos(M):
Right triangle with sides 8, 15, 17 and angle R at the bottom-left vertex
3.

Right triangle RSTRST has a right angle at TT, with RS=17RS = 17, RT=15RT = 15, and ST=8ST = 8. Which expression correctly gives cos⁡(R)\cos(R)?

4.

Triangle ABCABC has vertices with AC=10AC = 10, BC=7BC = 7, and AB=8AB = 8. None of the angles are labeled as 90°. Can a student apply SOH-CAH-TOA directly to find an unknown side?

5.

Right triangle UVWUVW has a right angle at WW, with VW=10VW = 10 and UW=5.77UW = 5.77. Compute tan⁡(U)\tan(U). Round to the nearest hundredth.

D

Word Problems

Draw a diagram, label the right triangle, and show your work.

Diagram of a ladder leaning against a wall, with the ladder at 35° to the ground
1.

A ladder 20 feet long leans against a vertical wall. The ladder makes a 35° angle with the ground.

How high up the wall does the ladder reach? Use sin⁡(35°)≈0.574\sin(35\degree) \approx 0.574. Round to the nearest hundredth.

Wheelchair ramp diagram with 2 ft rise and 20 ft horizontal run
2.

A wheelchair ramp rises 2 feet vertically over a horizontal distance of 20 feet.

1.

What is the length of the ramp surface (hypotenuse)? Use the Pythagorean theorem. Round to the nearest hundredth.

2.

What angle does the ramp make with the ground? Use the fact that tan⁡−1(0.1)≈5.71°\tan^{-1}(0.1) \approx 5.71\degree. State the angle.

Angle of elevation diagram: observer 100 m from building, 40° angle up to the top
3.

From a point 100 meters from the base of a building, the angle of elevation to the top of the building is 40°.

How tall is the building? Use tan⁡(40°)≈0.839\tan(40\degree) \approx 0.839. Round to the nearest hundredth.

E

Find the Mistake

Each problem shows student work that contains an error. Identify and explain the mistake.

1.

Jordan is working with two right triangles, both containing a 30° angle.

  • Small triangle: opposite = 3, hypotenuse = 6, so sin⁡(30°)=36=0.5\sin(30\degree) = \frac{3}{6} = 0.5
  • Large triangle: opposite = 9, hypotenuse = 18, so sin⁡(30°)=918\sin(30\degree) = \frac{9}{18}

Jordan concludes: "The large triangle gives a bigger ratio because it has larger sides."

What is Jordan's error?

Right triangle with sides 5, 12, 13 showing Priya's setup with the fraction 12/5 labeled incorrectly
2.

Priya sets up tan⁡(θ)\tan(\theta) for this right triangle: right angle at ZZ, with XY=13XY = 13, XZ=12XZ = 12, and YZ=5YZ = 5. Angle θ\theta is at vertex XX.

Priya writes: tan⁡(θ)=XZYZ=125\tan(\theta) = \dfrac{XZ}{YZ} = \dfrac{12}{5}

What error did Priya make?

3.

Amir has a right triangle where opposite = 7 and hypotenuse = 10. He wants to find angle θ\theta.

Amir writes: θ=sin⁡ ⁣(710)\theta = \sin\!\left(\dfrac{7}{10}\right)

What is wrong with Amir's equation?

F

Challenge Problems

These are extension problems. Show all work and reasoning.

Right triangle with angle A = 50° and AC = 15; side BC and hypotenuse AB are unknown
1.

Right triangle ABCABC has a right angle at CC. Angle A=50°A = 50\degree and AC=15AC = 15.

Find the length of side BCBC. Use tan⁡(50°)≈1.192\tan(50\degree) \approx 1.192. Round to the nearest hundredth.

2.

Explain why the similarity of right triangles is what makes trigonometric ratios well-defined functions of the angle. Address: (a) what would go wrong if similar right triangles did NOT have proportional sides, and (b) why the AA criterion applies to all right triangles with the same acute angle.

0 of 22 answered