Trigonometry Review and Practice Questions
- Posted by Brian Stocker MA
- Date June 20, 2014
- Comments 2 comments
Trigonometry is the branch of mathematics dealing with the relations of the sides and angles of triangles and with the relevant functions of any angles.
Quick Review of Concepts — Trigonometry Practice Questions — Answer Key — Common Trigonometry Mistakes
Quick Review with Examples
If we are observing a right triangle, where a and b are its legs and c is its hypotenuse, we can use trigonometric functions to make a relationship between angles and sides of the right triangle.
If the right angle of the right triangle ABC is at the point C, then the sine (sin) and the cosine (cos) of the angles α (at the point A) and β (at the point B) can be found like this:
sinα = a/c sinβ = b/c
cosα = b/c cosβ = a/c
Notice that sinα and cosβ are the equal, and same goes for sinβ and cosα. So, to find sine of the angle, we divide the side that is opposite of that angle and the hypotenuse. To find cosine of the angle, we divide the side that makes that angle (adjacent side) by the hypotenuse.
There are 2 more important trigonometric functions, tangent and cotangent:
tgα = sinα/cosα = a/b
ctgα = cosα/sinα = b/a
For the functions sine and cosine, there is a table with values for some of the angles, which is to be memorized as it is very useful for solving various trigonometric problems. Here is that table:
√
0⁰ | 30⁰ | 45⁰ | 60⁰ | 90⁰ | |
sinα | 0 | 1/2 | √2/2 | √3/2 | 1 |
cosα | 1 | √3/2 | √2/2 | 1/2 | 0 |
Let’s see one example:
If a is 9 cm and c is 18 cm, find α.
We can use the sine for this problem:
sinα = a/c = 9/18 = 1/2
We can see from the table that if sinα is 1/2, then angle α is 30⁰.
In addition to degrees we can write angles using π, where π represents 180⁰. For example, angle π/2 means a right angle of 90⁰.
Trigonometry Practice Questions
1. If sides a and b of a right triangle are 8 and 6, respectively, find cosine of α.
a. 1/5
b. 5/3
c. 3/5
d. 2/5
2. Find tangent of a right triangle, if a is 3 and c is 5.
a. 1/4
b. 5/3
c. 4/3
d. 3/4
3. If α=30^{0}, find sin30^{0} + cos60^{0}.
a. 1/2
b. 2/3
c. 1
d. 3/2
4. Calculate (sin^{2}30^{0 }– sin0^{0})/(cos90^{0 }– cos60^{0}).
A. -1/2
B. 2/3
C. 0
D. 1/2
5. Find cotangent of a right angle.
A. -1
B. 0
C. 1/2
D. -1/2
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Practice Question Answer Key
1. C 3/5
a = 8
b = 6
a^{2 }+ b^{2 }= c^{2}
8^{2 }+ 6^{2 }= c^{2}
64 + 36 = c^{2}
c^{2 }= 100
c = 10
cosα = b/c = 6/10 = 3/5
2. D 3/4
a = 3
c = 5
a^{2 } + b^{2 } = c^{2 }
3^{2 } + b^{2} = 5^{2}
b^{2} = 25 – 9
b^{2} = 16
b = 4
tgα = a/b = 3/4
3. C 1
α = 30^{0}
sin30^{0} + cos60^{0} = 1/2 + 1/2 = 1
4. A. -1/2
(sin^{2}30^{0}– sin0^{0}) / (cos90^{0} – cos60^{0}) = ((1/2)^{2 }– 0) / (0 – 1/2) = (1/4) / (-1/2) = -1/2
5. B. 0
α = 90^{0}
ctg90^{0} = cos90^{0}/ sin90^{0} = 0/1 = 0
Common Trigonometry Mistakes on a Test
Incorrect Use of Trigonometric Ratios
Mistake: Confusing sine, cosine, and tangent ratios when answering.
Explanation: Ratio represent a relationship between the sides of a right triangle.
Incorrect Mode (Degrees vs. Radians)
Mistake: Using the wrong mode.
Explanation: Trigonometric functions can be evaluated in degrees or radians.
Incorrect Signs in Quadrants
Mistake: Forgetting trigonometric functions have different signs in different quadrants.
Explanation: Use the ASTC rule (All Students Take Calculus) to remember which functions are positive in which quadrants.
Misidentifying Reference Angles
Mistake: Confusing reference angles.
Incorrect Use of Inverse Trigonometric Functions
Mistake: Confusing the results of inverse trig functions.
Explanation: The solution of an inverse trigonometric functions must be within the range.
Forgetting to Rationalize Denominators
Mistake: Leaving answers with irrational numbers in the denominator.
Explanation: Solutions should be in simplified form.
Incorrect Angle Measurement in Triangles:
Mistake: Not using the sum of angles in a triangle correctly.
Explanation: The sum of the angles in any triangle is always 180∘.
Not Checking Units
Mistake: Mixing up radians and degrees without proper conversion.
Explanation: Keep conversions consistent.
Not Simplifying Expressions
Mistake: Not simplifying answers.
Explanation: Answers are generally expected to be in the simplified form.
Date Published: Friday, June 20th, 2014
Date Modified: Thursday, June 6th, 2024
2 Comments
Nice questions will u plz add some more interesting questions from trigonometry
Nice and useful Trig work for Students’ review.