ACT study guide Math: Common Trigonometric Concepts Practice Questions

Recall that the tangent of an angle in a right triangle is the ratio of the side opposite the angle to the side adjacent to the angle. From the figure shown of △ABC, the side opposite angle C is 9 and the side adjacent to C is 5. Therefore tan C= 9/5

For right △ ABC with dimensions in centimeters as given below, what is tan C?

To find tan B in △ABC, take the ratio of the length of the opposite side to the length of the adjacent side: AC⎯⎯⎯⎯⎯ to AB⎯⎯⎯⎯⎯ = c to a, or c/a

For the right triangle △ABC shown below, what is tan B?

The tangent of an angle is the ratio of the side opposite the angle to the side adjacent to the angle. The side opposite angle x is 15 and the side adjacent is 8, making tan x=15/8.

For the triangle shown below, what is the value of tan x?

(By definition, sin x = opposite/hypotenuse, where "opposite" means the length of the side opposite the angle and "hypotenuse" is the length of the hypotenuse. Here, sin x= 9/15=) 3/5

For the triangle shown in the following figure, what is the value of sin x ?

(Given that tanx= 1/2 =opposite/adjacent and h = the length of the hypotenuse, then h2 = 12 + 22, and h=√5.Therefore, sinx=opposite/hypotenuse=1√15=)√5/5

Given the triangle in the following figure, if tanx=1/2, then sin x =

If cos A=45, and sin A=35, then tan A = ?

Because cos A=4/5, let the side adjacent to angle A be 4 and the hypotenuse be 5. Likewise, since sin A=3/5, let the side opposite angle A be 3 and the hypotenuse be 5. It follows then that tan A = side opposite divided by side adjacent, or 3/4

(In a right triangle, the sine of the angle x is the ratio of the side opposite the angle to the hypotenuse: sin(x)=20/40=0.5. (Note: If you're using a calculator, make sure you're in the degrees mode before taking the inverse sine.) x=sin−1(0.5)=)30°

What is the value of angle x in the diagram?(Figure is not to scale.)

The first step in solving this problem is to calculate the length of the hypotenuse. If you recognize that this is a special triangle, you know that the hypotenuse is 5. You could also let the answer choices guide you, since they each include some combination of 3, 4, and 5. Otherwise, use the Pythagorean Theorem, as follows: a2 + b2 = c2 32 + 42 = c2 9 + 16 = c2 25 = c2 c = 5 Next, recall that cosine is equal to the length of the adjacent side over the length of the hypotenuse. Therefore, cos ∠Z is 3/5.

For right triangle △XYZ below, what is cos ∠Z?

If sinA=35, then which of the following could be tan A?

Given that sinA=3/5, you might want to draw a picture like the one below to illustrate the situation. In this case, the ratio of the side opposite angle A to the hypotenuse is 3/5. To find tan A, the length of the other leg is needed. This can be obtained using the Pythagorean Theorem, or by knowing that this is a special 3-4-5 right triangle. According to the Pythagorean Theorem (in a right triangle with sides a, b, and c, where c is the hypotenuse, c2 = a2 + b2, 52 = 32 + x2, where x is the length of the unknown leg. Solve for x: x2 = 52 − 32 x2 = 25 − 9 x2 = 16 x = 4 Once the length of the third leg is obtained, tanA=opposite/adjacent=3/4.

To find the length of the segment AC⎯⎯⎯⎯⎯ in △ABC, where the length of the hypotenuse is 17, and the cosine of ∠C is 3/5 , use the definition of cosine: the ratio of the lengths of the adjacent side to the length of the hypotenuse. In △ABC, cosine of ∠ C is the ratio of the segment AC⎯⎯⎯⎯⎯ to the length of the hypotenuse. After substituting the length of the hypotenuse, we get 3/5=AC⎯⎯⎯⎯⎯/17 , and AC⎯⎯⎯⎯⎯=(17 × 3)/5, or 10.2 feet.

The hypotenuse of the right △ ABC shown below is 17 feet long. The cosine of angle C is 3/5. How many feet long is the segment AC⎯⎯⎯⎯⎯?

To solve this problem, let the height of the clock tower be x. Using the fact that the figure shown forms a right triangle with legs 150 and x, the height of the clock tower can be calculated using the trigonometric ratio tangent. Because the tangent of an angle is the ratio of the side opposite an angle to the side adjacent to the angle, tan 40°=x/150 . In the problem it is given that tan 40° ≈0.84, making 0.84=x/150, or x =150(0.84)=126.0. The height of the clock tower is therefore 126.0 feet.

A clock tower casts a 150-foot shadow on level ground, as shown below. The angle of elevation from the tip of the shadow to the top of the tower is 40°. To the nearest tenth of a foot, what is the height of the clock tower? (Note:cos 40∘=sin 50∘≈0.77 cos 50∘=sin 40∘≈0.64 tan 50∘≈1.19 tan 40∘≈0.84)

Tangent is the ratio of the side opposite to the side adjacent to an angle in a right triangle. Drawing a line that passes through (3, 3) and is perpendicular to the x-axis creates a right triangle, as shown in the figure (see below). Because point (3, 3) is given, both legs of the right triangle have a length of 3. Thus, tanφ=3/3=1

In the figure below, tan φ = ?

(Let w be the width of the rectangle and l be the length. We are given that x=8/3=lAE. From the figure, w = AE⎯⎯⎯⎯⎯ +2; substituting AE⎯⎯⎯⎯⎯ from the solution to tan x, 3 + 2 = 5. The area is thus w × l = 5 × 8 = )40.

In the following figure, ABCD is a rectangle such that tanx=8/3. In square feet, what is the area of the rectangle?

(The cosine of x is the ratio of the lengths of the side adjacent to the angle and the hypotenuse. Here, those sides have lengths b and a, respectively.) b/a

In the following figure, what is the value of cos x?

(Using the definition of sine as the opposite side divided by the hypotenuse, sin B=3/4=x1/2. Cross multiplying yields the equation 36 = 4x, so )x = 9

In the following figure, △ ABC is a right triangle such that the sine of ∠B is 34. What is the value of x?

Since sin Z is the ratio of the side opposite the angle to the hypotenuse, the side opposite Z, which is XY⎯⎯⎯⎯⎯ , must be determined. To do so, apply the Pythagorean Theorem, as follows: 102=XY⎯⎯⎯⎯⎯2+42XY⎯⎯⎯⎯⎯2=100−16=84XY⎯⎯⎯⎯⎯=84⎯⎯⎯⎯√=((21)(4))⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯√=21⎯⎯⎯⎯√4⎯⎯√, or 221⎯⎯⎯⎯√. Since sin Z is the ratio of the side opposite the angle to the hypotenuse, sin Z=22/1⎯⎯⎯⎯√10

In the right triangle below, YZ⎯⎯⎯⎯⎯ = 10 units, and XZ⎯⎯⎯⎯⎯ = 4 units. What is sin Z?

One way to solve this problem is to use SOH CAH TOA to find the sin, cos, and tan of ∠X, as follows: SOH: sin = opposite/hypotenuse = 12/13 CAH: cos = adjacent/hypotenuse = 5/13 TOA: tan = opposite/adjacent = 12/5 Only answer choice B, sin∠X=12/13, is a true statement about ∠X

In the right triangle shown below, which of the following statements is true about ∠X ?

(One way to solve this problem is to use SOH CAH TOA for angle A. You are given that the sine of angle A is 3/5, which means that the ratio of the length of the side opposite angle A to the length of the hypotenuse is 3 to 5. So, the ratio 3:5 is equal to the ratio BC⎯⎯⎯⎯⎯:AC⎯⎯⎯⎯⎯. You are given that AC⎯⎯⎯⎯⎯, the hypotenuse, is equal to 16, so now you can set up a proportion, cross-multiply, and solve for BC⎯⎯⎯⎯⎯ , as follows: 3/5=BC⎯⎯⎯⎯⎯/AC⎯⎯⎯⎯⎯5BC⎯⎯⎯⎯⎯= 3AC⎯⎯⎯⎯⎯5BC⎯⎯⎯⎯⎯= 3(16)5BC⎯⎯⎯⎯⎯= 48BC⎯⎯⎯⎯⎯= )9.6

The hypotenuse of right △ ABC shown below is 16 inches long. The sine of angle A is 3/5. About how many inches long is BC⎯⎯⎯⎯⎯?

To find the length of the segment LM⎯⎯⎯⎯⎯⎯ in △LMN, where the length of the hypotenuse is 22 and the cosine of angle L is 3/4, use the definition of cosine, which is the ratio of the length of the adjacent side to the length of the hypotenuse. In △LMN, the cosine of angle L is the ratio of the length of segment LM⎯⎯⎯⎯⎯⎯ to the length of the hypotenuse. Substitute the length of the hypotenuse and solve for LM⎯⎯⎯⎯⎯⎯ as follows: 34 = LM⎯⎯⎯⎯⎯⎯/22 4×LM⎯⎯⎯⎯⎯⎯ = 22×3 4×LM⎯⎯⎯⎯⎯⎯ = 66 LM⎯⎯⎯⎯⎯⎯= 66/4 LM⎯⎯⎯⎯⎯⎯ = 16.5

The hypotenuse of the right △LMN shown below is 22 feet long. The cosine of angle L is 3/4. How many feet long is the segment LM⎯⎯⎯⎯⎯⎯?

(To find cos π/12 using cos α−β=(cosα)(cosβ)+(sinα)(sinβ) given that π/12=π/3−π/4, you can first substitute π/3 for α and π/4 for β and get cos(π/3−π/4)=(cosπ/4)+(sinπ/3)(sinπ/4). Using the table of values to substitute into the equation, you get cosπ/12=(1/2)(√2/2)+(√3/2)(√2/2), or )(√6+√2)/4

What is cos π/12 given that π/12=π/3−π/4 and that cos(α − β) = (cos α) (cos β) + (sin α) (sin β)?

The sine of any acute angle is calculated by dividing the length of the side opposite to the acute angle by the hypotenuse (sin=opp/hyp). It may help you to draw a diagram to solve this problem, as shown below: Use the Pythagorean Theorem to calculate the length of the hypotenuse. a2 + b2 = c2 42 + 42 = c2 16 + 16 = c2 32 = c2 √32= c √16√2= c 4√2= c So, the sine of angle α is 4/4√2, which can be reduced to 1√2. Recall that roots should be eliminated from denominators: 1√2 (√2)/(√2)=⎯⎯√2/2.

In the figure below, sin α = ?

If cosx=57 and tanx=45, what is sin x?

The cosine of any acute angle is the length of the adjacent side divided by the hypotenuse (adj/hyp). The tangent of any acute angle is the length of the opposite side divided by the length of the adjacent side (opp/adj). The sine of any acute angle is the length of the opposite side divided by the hypotenuse. You are given that cosx=5/7(adj/hyp) and tanx=4/5(opp/adj). sin x must equal 4/7(opp/hyp).

((Remember: 1tan=cot)Tangent is the ratio of the side opposite to the side adjacent to an angle in a right triangle. If the distance, in feet, to the cell phone tower is x, then tan 41∘=200/x, or x=200/tan41∘. Since cot41∘=1 tan/41∘, x=200/tan41∘=)200cot41∘.

When measured from a point on the ground that is a certain distance from the base of a cell phone tower, the angle of elevation to the top of the tower is 41°, as shown below. The height of the cell phone tower is 200 feet. What is the distance, in feet, to the cell phone tower?