Precalculus Right Triangle and Law of Sines/Cosines Study Guide

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Q1. Find the exact value of each of the six trigonometric functions of the angle P.

Background

Topic: Right Triangle Trigonometry

This question tests your understanding of how to find the sine, cosine, tangent, cosecant, secant, and cotangent of a given angle in a right triangle.

Key Terms and Formulas:

Sine:
Cosine:
Tangent:
Cosecant:
Secant:
Cotangent:

Step-by-Step Guidance

Identify the side lengths of the triangle relative to angle (opposite, adjacent, hypotenuse).
Write the ratios for , , and using the side lengths.
Express , , and as the reciprocals of the primary functions.
Simplify each ratio to its lowest terms if possible.

Try solving on your own before revealing the answer!

Q2. Find when and in a right triangle.

Background

Topic: Right Triangle Trigonometry

This question asks you to find the cosine of an angle in a right triangle given two side lengths.

Key Terms and Formulas:

Cosine:
In a right triangle, and typically represent legs and hypotenuse, but check the triangle's labeling.

Step-by-Step Guidance

Determine which side is adjacent to angle and which is the hypotenuse.
Set up the ratio using the given values.
Simplify the fraction if possible.

Try solving on your own before revealing the answer!

Q3. Given , , find , , and in the right triangle.

Background

Topic: Solving Right Triangles

This question tests your ability to use trigonometric ratios and the Pythagorean theorem to find missing sides and angles in a right triangle.

Key Terms and Formulas:

Pythagorean Theorem:
Sine:
Cosine:
Tangent:

Step-by-Step Guidance

Use the given angle and side to find another side using a trigonometric ratio (e.g., or ).
Use the Pythagorean theorem to find the hypotenuse once you have both legs.
Find by subtracting from (since the triangle is right).

Try solving on your own before revealing the answer!

Q4. Given , , find , , and in the right triangle.

Background

Topic: Solving Right Triangles

This question asks you to find the hypotenuse and the two non-right angles of a right triangle given both legs.

Key Terms and Formulas:

Pythagorean Theorem:
Inverse Trig Functions: or

Step-by-Step Guidance

Calculate using the Pythagorean theorem.
Find using or .
Find by subtracting from .

Try solving on your own before revealing the answer!

Q5. Find the exact value of (do not use a calculator).

Background

Topic: Trigonometric Identities

This question tests your ability to use product-to-sum identities to simplify trigonometric expressions.

Key Terms and Formulas:

Product-to-Sum Identity:

Step-by-Step Guidance

Identify and .
Apply the product-to-sum identity to rewrite .
Simplify the resulting expression by calculating and .

Try solving on your own before revealing the answer!

Q6. Simplify (do not use a calculator).

Background

Topic: Trigonometric Identities and Simplification

This question tests your understanding of cofunction identities and simplifying trigonometric expressions.

Key Terms and Formulas:

Cofunction Identity:

Step-by-Step Guidance

Recognize that using the cofunction identity.
Rewrite in terms of .
Simplify the expression .

Try solving on your own before revealing the answer!

Q7. Simplify (do not use a calculator).

Background

Topic: Trigonometric Identities and Simplification

This question tests your ability to use cofunction identities and simplify trigonometric expressions.

Key Terms and Formulas:

Cofunction Identity:

Step-by-Step Guidance

Recognize that using the cofunction identity.
Rewrite in terms of .
Simplify the expression .

Try solving on your own before revealing the answer!

Q8. A 22-foot extension ladder leaning against a building makes a angle with the ground. How far up the building does the ladder touch?

Background

Topic: Right Triangle Applications

This question tests your ability to apply trigonometric ratios to solve real-world right triangle problems.

Key Terms and Formulas:

Sine:
Cosine:

Step-by-Step Guidance

Draw a right triangle representing the ladder, wall, and ground.
Identify the side you are solving for (vertical height up the building = opposite side).
Set up the equation using .
Rearrange to solve for the opposite side.

Try solving on your own before revealing the answer!

Q9. A 12 meter flagpole casts a 9 meter shadow. Find the angle of elevation to the sun from the ground.

Background

Topic: Right Triangle Applications

This question tests your ability to use trigonometric ratios to find an angle given two sides of a right triangle.

Key Terms and Formulas:

Tangent:
Inverse Tangent:

Step-by-Step Guidance

Identify the opposite side (flagpole height) and adjacent side (shadow length).
Set up the equation .
Take the inverse tangent to solve for .

Try solving on your own before revealing the answer!

Q10. You’ve let out all 100 feet of string for your kite, and the angle that the string makes with the ground is . How tall is the tree?

Background

Topic: Right Triangle Applications

This question tests your ability to use trigonometric ratios to find the height of an object given the hypotenuse and an angle.

Key Terms and Formulas:

Sine:

Step-by-Step Guidance

Draw a right triangle with the string as the hypotenuse and the height of the tree as the opposite side.
Set up the equation .
Rearrange to solve for the height.

Try solving on your own before revealing the answer!

Q11. Solve the triangle(s) given , , .

Background

Topic: Law of Sines (SAA/ASA/SSA Triangles)

This question tests your ability to solve a triangle using the Law of Sines when two angles and a side are given.

Key Terms and Formulas:

Law of Sines:

Step-by-Step Guidance

Find the missing angle using the triangle sum theorem: .
Set up the Law of Sines to solve for the missing sides and .
Plug in the known values and solve for one missing side at a time.

Try solving on your own before revealing the answer!

Q12. Solve the triangle(s) given , , .

Background

Topic: Law of Sines (SSA Triangles)

This question tests your ability to solve a triangle using the Law of Sines when two sides and a non-included angle are given (SSA case).

Key Terms and Formulas:

Law of Sines:

Step-by-Step Guidance

Set up the Law of Sines to solve for one of the unknown angles (e.g., ).
Use the triangle sum theorem to find the remaining angle.
Solve for the missing side using the Law of Sines.

Try solving on your own before revealing the answer!

Q13. Solve the triangle(s) given , , .

Background

Topic: Law of Sines (SSA Triangles)

This question tests your ability to solve a triangle using the Law of Sines when two sides and a non-included angle are given (SSA case).

Key Terms and Formulas:

Law of Sines:

Step-by-Step Guidance

Set up the Law of Sines to solve for .
Use the triangle sum theorem to find .
Solve for the missing side using the Law of Sines.

Try solving on your own before revealing the answer!

Q14. Find the missing part of the triangle(s): , , .

Background

Topic: Law of Sines (SAA/ASA Triangles)

This question tests your ability to solve a triangle using the Law of Sines when two angles and a side are given.

Key Terms and Formulas:

Law of Sines:

Step-by-Step Guidance

Find the missing angle using the triangle sum theorem.
Set up the Law of Sines to solve for the missing sides and .
Plug in the known values and solve for one missing side at a time.

Try solving on your own before revealing the answer!

Q15. Find the missing part of the triangle(s): , , .

Background

Topic: Law of Sines (SSA Triangles)

This question tests your ability to solve a triangle using the Law of Sines when two sides and a non-included angle are given (SSA case).

Key Terms and Formulas:

Law of Sines:

Step-by-Step Guidance

Set up the Law of Sines to solve for one of the unknown angles (e.g., or ).
Use the triangle sum theorem to find the remaining angle.
Solve for the missing side using the Law of Sines.

Try solving on your own before revealing the answer!

Q16. John wants to measure the height of a tree. He walks exactly 100 feet from the base of the tree and looks up. The angle from the ground to the top of the tree is . This particular tree grows at an angle of with respect to the ground rather than vertically. How tall is the tree?

Background

Topic: Law of Sines (Oblique Triangle Application)

This question tests your ability to apply the Law of Sines to solve for the height of an object when the triangle is not a right triangle.

Key Terms and Formulas:

Law of Sines:

Step-by-Step Guidance

Draw the triangle and label all known sides and angles.
Use the triangle sum theorem to find the missing angle.
Set up the Law of Sines to solve for the side representing the height of the tree.

Try solving on your own before revealing the answer!

Q17. Solve the triangle given , , .

Background

Topic: Law of Cosines (SAS/SSS Triangles)

This question tests your ability to use the Law of Cosines to solve a triangle when two sides and the included angle are given (SAS case).

Key Terms and Formulas:

Law of Cosines:

Step-by-Step Guidance

Identify which side and angle are opposite each other.
Use the Law of Cosines to solve for the missing side.
Use the Law of Sines or Cosines to find the remaining angles.

Try solving on your own before revealing the answer!

Q18. Solve the triangle given , , .

Background

Topic: Law of Cosines (SSS Triangles)

This question tests your ability to use the Law of Cosines to solve a triangle when all three sides are given (SSS case).

Key Terms and Formulas:

Law of Cosines:
Law of Sines:

Step-by-Step Guidance

Use the Law of Cosines to solve for one angle (e.g., ).
Use the Law of Sines to solve for another angle.
Find the third angle using the triangle sum theorem.

Try solving on your own before revealing the answer!

Q19. Solve the triangle given , , .

Background

Topic: Law of Cosines (SAS Triangles)

This question tests your ability to use the Law of Cosines to solve a triangle when two sides and the included angle are given (SAS case).

Key Terms and Formulas:

Law of Cosines:
Law of Sines:

Step-by-Step Guidance

Use the Law of Cosines to solve for the missing side .
Use the Law of Sines to solve for one of the remaining angles.
Find the third angle using the triangle sum theorem.

Try solving on your own before revealing the answer!

Q20. Find the missing part of the triangle(s): , , .

Background

Topic: Law of Cosines (SSS Triangles)

This question tests your ability to use the Law of Cosines to solve a triangle when all three sides are given (SSS case).

Key Terms and Formulas:

Law of Cosines:
Law of Sines:

Step-by-Step Guidance

Use the Law of Cosines to solve for one angle (e.g., ).
Use the Law of Sines to solve for another angle.
Find the third angle using the triangle sum theorem.

Try solving on your own before revealing the answer!

Q21. The distance from home plate to dead center field in US Cellular Field is 400 feet. A baseball diamond is a square with a distance from home plate to first base of 90 feet. How far is it from first base to dead center field?

Background

Topic: Law of Cosines (Application Problem)

This question tests your ability to model a real-world situation with a triangle and use the Law of Cosines to find an unknown distance.

Key Terms and Formulas:

Law of Cosines:

Step-by-Step Guidance

Draw a diagram of the baseball field and label all known distances.
Identify the triangle formed by home plate, first base, and dead center field.
Determine the angle at first base (likely since the diamond is a square).
Set up the Law of Cosines to solve for the unknown distance from first base to dead center field.