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Here are a few recommended readings before getting started.
Try your knowledge on these topics.
The Leaning Tower of Pisa has a tilt of 4 degrees. Once, a worker maintaining it accidentally dropped a hammer from the top. The hammer landed 3 meters away from the base of the tower. Luckily, it did not hurt anyone!
Because all right angles are congruent, all right triangles have one pair of congruent angles. If they also have one pair of congruent acute angles, then the triangles have two pairs of congruent angles. Therefore, by the Angle-Angle Similarity Theorem, two triangles with one pair of congruent acute angles are similar.
Since corresponding sides of similar polygons are proportional, the ratios between corresponding sides of similar right triangles are the same.
The ratios between side lengths of right triangles depend on the acute angles of the triangle. Some of these ratios receive a special name.
A trigonometric ratio relates two side lengths of a right triangle. Consider the right triangle △ABC. One of its acute angles has been named θ.
Since it is opposite to the right angle, BC is the hypotenuse of the right triangle. The remaining sides — the legs — can be named relative to the marked angle θ. Because AB is next to ∠θ, it is called the adjacent side. Conversely, because AC lies across from ∠θ, it is called the opposite side.
The names of the three main ratios between side lengths are stated in the following table.
Name | Definition | Notation |
---|---|---|
Sine of ∠θ | HypotenuseLength of opposite side to ∠θ | sinθ=hypopp |
Cosine of ∠θ | HypotenuseLength of adjacent side to ∠θ | cosθ=hypadj |
Tangent of ∠θ | Length of adjacent side to ∠θLength of opposite side to ∠θ | tanθ=adjopp |
Dominika is helping Tadeo understand trigonometric ratios. She drew three right triangles for him to write trigonometric ratios with respect to the acute angle θ. Help Tadeo grasp this topic by selecting the correct answers!
Despite the awesome explanations Dominika provided, Tadeo still does not get how to find trigonometric ratios. To help his friend, Dominika thought of one more exercise.
This time, Dominika drew one right triangle and stated its three side lengths. She also labeled one of the triangle's acute angles.
Tadeo has now been asked to find the sine, cosine, and tangent of ∠θ. Help him find the answers!Start by identifying the hypotenuse of the right triangle. Then identify the opposite and adjacent sides to ∠θ. Finally, recall the definitions of sine, cosine, and tangent of an acute angle of a right triangle.
Definition | Substitute |
---|---|
sinθ=HypotenuseLength of opposite side to ∠θ | sinθ=178 |
cosθ=HypotenuseLength of adjacent side to ∠θ | cosθ=1715 |
tanθ=Length of adjacent side to ∠θLength of opposite side to ∠θ | tanθ=158 |
Tadeo finally understands the topic! But wait, Dominika wants to level up and has let him know that trigonometric ratios can also be used to find missing side lengths of a right triangle. "Tell me more," Tadeo responds. An acute angle and the hypotenuse of a right triangle are given. To see whether Tadeo masters this topic, Dominika asked him to find the value of x, which is the length of the opposite side to the given angle.
Help Tadeo find the value of x. If necessary, round the answer to three significant figures.Identify the trigonometric ratio that should be used according to the given and desired lengths. Then, with the help of a calculator, set and solve an equation.
The hypotenuse of the right triangle is given, and the length of the opposite side to the given angle is to be found.
The trigonometric ratio that relates these two sides and the acute angle is the sine ratio.LHS⋅10=RHS⋅10
10a⋅10=a
Multiply
Rearrange equation
Degreein the third row.
Next the value of sin60∘ can be calculated by pushing SIN followed by the angle measure.
Use a calculator
Multiply
Round to 3 significant digit(s)
In the right triangles below, one acute angle and one side length are given. By using the corresponding trigonometric ratio, find the length of the side labeled x. Round the answer to one decimal place.
By using trigonometric ratios, an important property of angles can be derived.
For any angle θ, the following trigonometric identities hold true.
Definition | Substitute | Simplify | |
---|---|---|---|
sinθ | HypotenuseLength of opposite side to ∠θ | 1opp | opp |
cosθ | HypotenuseLength of adjacent side to ∠θ | 1adj | adj |
It can be seen that if the hypotenuse of a right triangle is 1, the sine of an acute angle is equal to the length of its opposite side. Similarly, the cosine of the angle is equal to the length of its adjacent side.
By the Pythagorean Theorem, the sum of the squares of the legs of a right triangle is equal to the square of the hypotenuse. Therefore, for the above triangle, the sum of the squares of sinθ and cosθ is equal to the square of 1.
The property seen before can be used, among other things, to find the sine or cosine ratio of an acute angle in a right triangle.
Kriz and his friends plan to spend Saturday afternoon playing video games. To optimize the space, they decide to tidy up the basement to ensure the console, snacks, and beverages are placed in the form of a right triangle. Kriz decides to set the snacks and the beverages 3 and 5 meters away from the console, respectively.The hypotenuse of the right triangle is 5 and the measure of the opposite side to ∠θ is 3. With this information, the sine ratio can be found.
Trigonometric ratios can also be used to find missing angles. Consider a right triangle △ABC where the hypotenuse and a leg are given.
Suppose now that the measure of ∠C is desired. Note that, apart from the hypotenuse, the side whose length is known is opposite to ∠C. The trigonometric ratio that relates the hypotenuse and the opposite side to an acute angle in a right triangle is the sine ratio.Previously, it was said that apart from being useful to find side lengths of a right triangle, trigonometric ratios can also be used to find missing angle measures.
Before playing video games with his friends, Kriz wants to finish his math homework to have a care-free weekend. He wants to find the measure of an acute angle in three different right triangles. By using the corresponding trigonometric ratios, help Kriz find m∠θ in each triangle. Round the answer to the nearest degree.
Degreein the third row.
Next the value of tan-11235, can be calculated by pushing 2ND, followed by TAN, and 35/12.
Degreein the third row.
Next the value of cos-1257, is calculated by pushing 2ND, followed by COS, and 7/25.
Degreein the third row.
Next the value of sin-12920, can be calculated by pushing 2ND, followed by SIN, and 20/29.
In the following right triangles, two side lengths are given. By using the corresponding trigonometric ratio, find m∠θ. Round the answer to nearest degree.
Apart from the sine, cosine, and tangent ratios, there are three other trigonometric ratios that are worth mentioning.
Consider the right triangle △ABC.
The so called reciprocal ratios are written in the next table.
Name | Definition | Notation |
---|---|---|
Cosecant of ∠θ | Length of opposite side to ∠θHypotenuse | cscθ=opphyp |
Secant of ∠θ | Length of adjacent side to ∠θHypotenuse | secθ=adjhyp |
Cotangent of ∠θ | Length of opposite side to ∠θLength of adjacent side to ∠θ | cotθ=oppadj |
These ratios can be defined in terms of sine, cosine, and tangent.
The trigonometric ratios cosecant, secant, and cotangent are reciprocals of sine, cosine, and tangent, respectively.
cscθ=sinθ1
secθ=cosθ1
cotθ=tanθ1
Consider a right triangle with the three sides labeled with respect to an acute angle θ.
Next, the sine, cosine, tangent, cosecant, secant, and cotangent ratios are written.LHS⋅sinθ1=RHS⋅sinθ1
LHS⋅opphyp=RHS⋅opphyp
Rearrange equation
If the sine, cosine, and tangent ratios are known, then their reciprocals cosecant, secant, and cotangent can be calculated without too much effort.
LaShay is really good at her favorite subject, Geometry. She has been appointed by Jefferson High's principal to do some tutoring for some of her classmates after school. To do so, she drew a right triangle. She then asked her peers to find all six trigonometric ratios with respect to the marked angle θ.
Help LaShay's classmates find the trigonometric ratios!
Identify the hypotenuse of the right triangle and the opposite and adjacent sides to ∠θ.
The hypotenuse of the right triangle and the opposite and adjacent sides to ∠θ will be identified.
It can be seen that the hypotenuse is 101 and the lengths of the opposite and adjacent sides to ∠θ are 20 and 99, respectively. With this information, the sine, cosine, and tangent ratios can be found.